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New exergy analysis of a regenerative closed Brayton cycle
Energy Conversion and Management 134 (2017) 116–124
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
New exergy analysis of a regenerative closed 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 10 August 2016 Received in revised form 14 November 2016 Accepted 10 December 2016
Keywords: Power Regenerative Brayton cycle Exergy Efficiency Optimization
a b s t r a c t In this study, the optimal performance of a regenerative closed Brayton cycle is sought through power maximization. Optimization is performed on the output power as the objective function using genetic algorithm. In order to take into account the time and the size constraints in current problem, the dimensionless mass-flow parameter is used. The influence of the unavoidable exergy destruction due to finitetime constraint is taken into account by developing the definition of heat exergy. Finally, the improved definitions are proposed for heat exergy, and the second law efficiency. Moreover, the new definitions will be compared with the conventional ones. For example, at a specified dimensionless mass-flow parameter, exergy overestimation in conventional definition, causes about 31% lower estimation of the second law efficiency. These results could be expected to be utilized in future solar thermal Brayton cycle assessment and optimization. Ó 2016 Elsevier Ltd. All rights reserved.
0. Introduction The underlying functional point of cyclic heat engines is maximum power state. The operational point in maximum power and reversible performance possess the same importance. The efficiency of heat engines are restricted by Carnot efficiency. This efficiency is obtainable in the reversible case. Practically, all thermodynamic processes take place in finite-size components during finite-time, which leads to irreversibility (exergy destruction). Accordingly, while Carnot cycle gives upper bound for thermal efficiency, it cannot be a comparison standard for real heat engines. Analysis techniques have been developed in various studies to consider the internal and/or external irreversibility in heat engines. Curzon and Ahlborn [1] studied the effect of external irreversibility, which accounted for irreversibilities in the heatexchange processes between the power cycle and its heat sources, on Carnot cycle’s output power and thermal efficiency. This system was entitled endoreversible due to internal reversibility of cycle. In their research, thermal efficiency at the maximum power state was expressed in the form of Eq. (1). T L and T H are temperatures at cold and hot heat exchangers, respectively.
gCA
sffiffiffiffiffiffi TL ¼1 TH
⇑ Corresponding author. E-mail address: [email protected] (S. Farahat). http://dx.doi.org/10.1016/j.enconman.2016.12.020 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.
ð1Þ
Bejan [2] showed that the degree of thermodynamic imperfection of power plants could be estimated based on a very simple model that considers only the sources of heat transfer irreversibilities. In a separate study, Bejan [3] investigated the optimal allocation of heat exchange equipment. His study showed that the power output of various power plant configurations could be maximized by properly dividing the fixed inventory of heat exchange equipment among the heat transfer components of each plant. Wu [4] established a comparison between endoreversible Carnot cycle and the same system with both internal and external irreversibility. It was shown that the internal irreversibility reduces power and efficiency. Gordon [5] analyzed heat engines considering finite rate heat transfer and finite-capacity thermal reservoirs. He showed that the efficiency at maximum power depends on the thermal reservoir temperatures, and other system variables such as reservoir capacity or working fluid specific heats. Heat engines operate in finite time; therefore, the realistic study of their optimal performance is feasible through the concept of finite-time thermodynamics [6]. This method was applied to the optimization of regenerative endoreversible Brayton cycle with finite thermal capacitance rates in heat reservoirs [8]. In the undertaken research, application of regenerators led to decrease in the maximum power and thermal efficiency. Their study showed the regenerative heat-transfer rate was positive for low temperature ratios and negative for high temperature ratios. Further analyses were performed on regenerative and irreversible models of Brayton heat engines [9,10].
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Nomenclature T _ W S_ r E_ e cP P E_ _ m Q_ A U F R PR
temperature (K) power (kW) entropy rate (kW/K) thermal conductance ratio 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/m 2K) dimensionless mass flow rate specific gas constant (J/kg K) pressure ratio
Greek symbols g efficiency e effectiveness
Optimization of real systems is confined to thermal performance and physical constraints. Bejan [11] established two optimization approaches based on these 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. [13] used heat exchanger size as design constraints for irreversible regenerative Brayton cycle optimization. In this model, finite-time thermodynamics and optimization were used to determine the maximum power and minimum entropy generation, along with the global maximum net power. In various research studies different thermal parameters were used for thermodynamic optimization of heat engines. Power density and exergy density are two of these objective functions, applied to irreversible Brayton cycle with regeneration, and cogeneration system [12,14]. Parametric design for the maximum exergy density leads to a smaller and more efficient cogeneration system. Yang et al. [15] showed that regeneration and intercooling process can improve the economic profit rate and exergy efficiency of cogeneration plants. Haesli [16] has studied second law efficiency optimization of a regenerative Brayton cycle. He examined the maximum limit of the thermal efficiency of an ideal Brayton cycle, and defined the second law efficiency of the cycle as the ratio of the thermal (first law) efficiency to the maximum attainable efficiency. Tsatsaronis and Park [17] have divided exergy destruction into avoidable and unavoidable parts and demonstrated how to estimate the avoidable and unavoidable exergy destruction associated with system’s components. F. Petrakopoulou et al. [18] analyzed a combined cycle power plant using both conventional and advanced exergetic analyses. They found that most of the exergy destruction in the plant components was unavoidable. Vucˇkovic´ et al. [19] investigated an industrial plant using both conventional and advanced exergy analysis. They claimed that highest exergy destruction was caused by the steam boiler. Moreover 92.34% of the total exergy destruction in boiler was unavoidable. Açıkkalp et al. [20] have analyzed a trigeneration system using an advanced exergy analysis. The results of their research indicated that the improvement potential of their system was low because 82% of the total exergy destruction cost rates were unavoidable. Naserian et al. sought the optimal performance of a regenerative Brayton
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 LTHE low temperature heat exchanger AC air compressor R heat regenerator HTHE high temperature heat exchanger
cycle through multi-objective ecological function maximization [21], and exergoeconomic multi objective optimization [22] using finite-time thermodynamic concept and finite-size components. Goodarzi [23] introduced and then analyzed energetically a new regenerative Brayton cycle. In current study, an irreversible closed-cycle Brayton engine with regeneration is investigated. Use of a closed-cycle system permits working fluids other than air, such as carbon dioxide [24], Helium [25]. Traditional Brayton cycles can be upgraded to supercritical CO2 to enable much greater efficiencies and power outputs [26]. Many studies have been published on the performance and optimization of the Brayton cycle and solar thermal Brayton cycle [27–32]. 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]. The main originality of the current research is as follows: The maximum attainable dimensionless net output power is obtained considering the impact of mass-flow parameter variation. The influence of dimensionless mass-flow parameter on internal, and external exergy destructions, and 1st and 2nd law efficiencies are investigated. The definitions of heat exergy, and second law efficiency are modified. 1. Heat engine model The schematic of the system, an irreversible regenerative closed Brayton cycle, is shown in Fig. 1. The heat engine includes an irreversible compressor, a regenerator, two heat exchangers, and an irreversible turbine. Heat exchangers are used to transfer heat from high temperature flow to the cycle and from the cycle to the low temperature flow. The entire analysis of system can be broken down into: (i) energy analysis (ii) exergy analysis. 1.1. Energy analysis The following assumptions are taken into account to develop thermodynamic model: All processes are steady state. Compressor
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Fig. 1. The schematic of the system.
and turbine are adiabatic. The efficiencies of the compressor and the turbine are considered 0.9 and 0.95, respectively. The temperature of working flow is variable. The chemical exergetic term does not vary. The kinetic and potential exergetic terms are neglected. The temperature of H1 and L1 flows are assumed 1200 K and 300 K, respectively. The dead-state condition is considered P0 ¼ 1:01 bar and T0 ¼ 293:15 K. The flowing gas is considered as a product of natural gas combustion process and it follows the principle of ideal-gas mixture. The heat capacity of the gas at constant pressure CP;gas varies with temperature as indicated in Eq. (2) [38].
cP;gas ðTÞ ¼ 0:93750 þ
0:01215 102
Tþ
0:01670 105
T2
0:07164 109
T3
_ m ðU:AÞT cP;min
i;0 < F < 1
ð3Þ
where cP;min is the least heat capacity of the gas in the heat engine. _ ¼ F:ððU:AÞT =cP;min Þ. Therefore, In the forthcoming equations m Energy balance equations for compressor (Eq. (4)), turbine (Eq. (5)), high temperature heat exchanger (Eq. (6)) and low temperature heat exchanger (Eq. (7)) are derived (in [21]) as follows.
8 <
39 cg 1 = 1 4 P 2 cg T2 ¼ T1 1 þ 15 : ; gc P1 8 <
_ P;LTHE min ðT 6 T L1 Þ ¼ eLTHE m:c
cP;HTHE min ¼ min cP;H ; cP;W1
ð8Þ
cP;LTHE min ¼ min cP;L ; cP;W2
ð9Þ
T H1 þ T 8 ¼ cP;gas 2
ð10Þ
cP;H
cP;L ¼ cP;gas
2
P4 T 5 ¼ T 4 1 þ ggt 41 : P5
39 1c cg = g 5 ;
ð5Þ
_ P;H ðT H1 T 8 Þ ¼ m:c _ P;W1 ðT 4 T 3 Þ Q_ H ¼ m:c _ P;HTHE min ðT H1 T 3 Þ ¼ eHTHE m:c
T L1 þ T 7 2
ð11Þ
cP;W1 ¼ cP;gas
T3 þ T4 2
ð12Þ
cP;W2 ¼ cP;gas
T6 þ T1 2
ð13Þ
Energy analysis of regenerator is studied in Eqs. (14)–(17).
_ P;R1 ðT 3 T 2 Þ ¼ mc _ P;R2 ðT 5 T 6 Þ Q_ R ¼ mc
eR ¼
T3 T2 T5 T2
ð14Þ ð15Þ
cP;R1 ¼ cP;g
T3 þ T2 2
ð16Þ
cP;R2 ¼ cP;g
T6 þ T1 2
ð17Þ
2
ð4Þ
ð7Þ
where the embedded parameters in above relations are described in Eqs. (8)–(13).
ð2Þ
In order to enter the size constraint to the analysis, total thermal conductance of system heat exchangers (ðU:AÞT ) is assumed to be constant [2,13,21,27,28]. Moreover, time will be constrained if the mass flow rate is finite and non-zero. For taking into account the time and the size constraints in current investigation, the dimensionless mass-flow parameter (F) is used. This parameter was defined by Naserian et al. [21], as Eq. (3).
F¼h
_ P;L ðT 7 T L1 Þ ¼ m:c _ P;W2 ðT 6 T 1 Þ Q_ L ¼ m:c
Finally, the net power in the heat engine is presented in Eqs. (18) and (19).
_ net ¼ Q_ H Q_ L W
ð18Þ
_ net ¼ eHTHE :m:c _ P;HTHE min :ðT H1 T 3 Þ eLTHE :m:c _ P;LTHE min :ðT 6 T L1 Þ W ð6Þ
ð19Þ
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1.2. Exergy analysis of system Exergy analysis helps to investigate the efficient part of energy at thermodynamic processes and the results of such analysis lead to operational improvements. In order to carry out an exergy analysis, an exergy model must be provided. Physical exergy for temperature dependent specific heat is defined as Eq. (20).
Z _ E_ ¼ m
Z
T
T
cP ðTÞdT T 0
T0
cP ðTÞ T0
dT P R ln T P0
ð20Þ
Physical exergy of flows (based on Eq. (2)) is expressed in Eqs. (21)–(25). For i ¼ H1; L1; 1; 2; 3; 4; 5; 6; 7; 8 02
_B E_ i ¼ m @4
1 3T i T 2 þ 0:00567 T 3 0:01791 T4 c0:93750T þ 0:006075 P 105 102 109 i 5 þ T 0 Rln C A 0:00835 2 0:02388 3 P0 T 0 0:93750lnðTÞ þ 0:01215 T þ T T 2 5 9 10 10 10 T0
ð21Þ
_ P;AC ðT 2 T 1 Þ E_ 9 ¼ mc
ð22Þ
_ P;gt ðT 4 T 5 Þ E_ 10 ¼ mc
ð23Þ
T4 þ T5 2
ð24Þ
T1 þ T2 ¼ cP;g 2
ð25Þ
cP;gt ¼ cP;g
cP;AC
1.2.1. Exergy of cycle components The fuel exergy rate (physical exergy related to the streams entering each component) and product exergy rate (physical exergy related to the streams exiting each component) for system components are tabulated in Table 1. Exergy destructions of system components are formulated in Eqs. (26)–(31) [7].
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, exergy, first law, and second law efficiencies corresponding to F, are going to be plotted. Finally, regarding the results, new definition of exergy, and second law efficiency will be obtained. 2. Optimization study For certain values of F, the system is simulated in MATLAB.. Outcome of this modeling is used in genetic algorithm for optimization purpose. Genetic algorithm has been used by several researchers (such as Naserian et al. [21,22], Ahmadi et al. [25], and Ahmadi et al. [29]) to optimize the operation of thermodynamic cycles. The flowchart of optimization process is shown in Fig. 2. The net output power of system is specified as the objective function for optimization (Eq. (33)).
_ ¼ E_ 10 E_ 9 W
ð33Þ
Maximize Eq. (33) The decision variables for optimization of the system are the compressor pressure ratio, and thermal conductance ratios of heat exchangers (Eqs. (34) and (35).
rHTHE ¼
ðU:AÞHTHE ðU:AÞT
ð34Þ
rLTHE ¼
ðU:AÞLTHE ðU:AÞT
ð35Þ
2.1. Constraints The significant constraint taken into account in this study is the heat exchangers’ size. The total thermal conductance of heat exchangers is constant, which results in Eq. (35), where r R is described as Eq. (37). The decision variables bounds, and the stopping criteria used for optimization problem are shown in Tables 3 and 4, respectively.
E_ D;k ¼ E_ F;k E_ P;k
ð26Þ
E_ D;1 ¼ E_ F;1 E_ P;1 ¼ E_ 9 ðE_ 2 E_ 1 Þ
ð27Þ
E_ D;2 ¼ E_ F;2 E_ P;2 ¼ ðE_ 5 E_ 6 Þ ðE_ 3 E_ 2 Þ
ð28Þ
E_ D;3 ¼ E_ F;3 E_ P;3 ¼ ðE_ H1 E_ 8 Þ ðE_ 4 E_ 3 Þ
ð29Þ
rR ¼
E_ D;4 ¼ E_ F;4 E_ P;4 ¼ ðE_ 4 E_ 5 Þ E_ 10
ð30Þ
The tuning parameters values for Genetic algorithm optimization are given in Table 2.
E_ D;5 ¼ E_ F;5 E_ P;5 ¼ ðE_ 6 E_ 1 Þ ðE_ 7 E_ L1 Þ
ð31Þ
ðU:AÞTOT ¼ Constant
ð36Þ
rHTHE þ rLTHE þ r R ¼ 1 0 < r < 1
ð37Þ
ðU:AÞR ðU:AÞT
ð38Þ
3. Results and discussion
The total exergy destruction of cycle is calculated in Eq. (32).
E_ D;T ¼
5 X E_ D;k
ð32Þ
k¼1
Table 1 Fuel and product exergy rates 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
Variation of dimensionless net power with parameter F is depicted in Fig. 3 at the maximum power state. Hereafter, all results are investigated at the maximum state even though it is not stated. The slope of power increases gradually with F up to the maximum of maximum power value at F = 0.3. Afterwards, the dimensionless net power slopes down sharply. Therefore, the maximum of maximum net power is located between the two limits. Exergy of high temperature thermal source, dimensionless input heat to the system ðQ_ H =ððU:AÞ T0 ÞÞ, dimensionless fuel exergy T
(input thermal exergy to the system) and dimensionless exergy of input heat to the system (exergy of fuel heat) are studied in Fig. 4 as a function of F. These parameters proportionally increase with the increase in F. Hot line exergy partially enters the system through high temperature heat exchanger and the rest exists within
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M.M. Naserian et al. / Energy Conversion and Management 134 (2017) 116–124 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 3 The bounds used for decision variables. Decision variable
Range
Pressure ratio of compressor r HTHE rLTHE
[1,20] [0,1] [0,1]
Table 4 Stop criteria for the optimization algorithms. Stop criterion
Value
Number of generation Function tolerance
1000 1e7
Fig. 3. Effect of F on the value of dimensionless maximum net power.
dimensionless fuel exergy on dimensionless exergy of fuel heat verifies the accuracy of energy and exergy calculations in current work.
3.1. Modification of heat exergy, and second law efficiency definitions
Fig. 2. The flowchart of optimization process.
fluid flow (the main objective of this study is to investigate input and output exergies of the cycle and their dependency on F; therefore, methods of using flow output exergy is not detailed here). Dimensionless of Q_ H , fuel exergy (E_ H1 E_ 8 ) and exergy of fuel heat (Q_ H ð1 T0 =TH Þ) behave similarly. The exact superimposition of
A system delivers the maximum possible work as it undergoes a reversible process from the specified initial state to the dead state. This represents the useful work potential of the system at the specified state and is called exergy. It is important to realize that exergy does not represent the amount of work that a work-producing device will actually deliver. There will always be a difference between exergy and the actual work delivered by a device. Irreversibilities such as friction, heat transfer through a finite temperature difference, nonquasiequilibrium compression or expansion and etc. always generate entropy, and anything that generates entropy always causes (the difference) exergy destruction [24]. Fig. 5 shows the changes in dimensionless total exergy destruction, internal exergy destruction ðE_ D;1 þ E_ D;2 þ E_ D;4 Þ, and external exergy
M.M. Naserian et al. / Energy Conversion and Management 134 (2017) 116–124
121
optimization [33]. Size and time are the sources of entropy generation in practical situations [21]. One of the most important disadvantages of conventional definition of exergy is that it is overestimate the power of actual systems. Therefore, the conventional definition of the second law efficiency of the system is lower estimated. The unavoidable part of exergy destruction (even if with technology improvement), never can be overcome. In order to avoid these exergy destructions, the reasonable approach is to express exergy as a function of size and time. The influence of exergy destruction due to finite-time constraint is taken into account by eliminating the exergy destructions of thermal sources, which are unavoidable, from conventional exergy definition as Eq. (38).
E_ F;New ¼ E_ F E_ D;HeatSources
Fig. 4. Effect of F on dimensionless Hot line exergy, Fuel exergy, Exergy of fuel heat, and dimensionless input heat to the system.
destruction ðE_ D;3 þ E_ D;5 Þ of the cycle, relating to F. According to what is observed, all of the parameters are equal to zero for the lowest value of F, as the external exergy destruction and internal exergy destruction are equal to zero. Hence, the efficiency approximates the Carnot efficiency as the process time approaches infinity. The parameters increase with increment of F. Comparison of the diagrams proves that a great part of total exergy destruction is derived from external exergy destruction of the system (i.e. the heat exchangers of thermal sources). Nevertheless, due to significant increment of exergy destruction in thermal sources, exergy destruction increases progressively and it reaches the maximum value at F = 1. Carnot cycle gives upper bound for thermal efficiency. However, to achieve this theoretical efficiency, the heat transfer process have to be carried out infinitely slowly (in infinite time) or in infinitely large reservoirs (in infinite size) so that the working substance can come into thermal equilibrium with the heat sources. Under these conditions the power output is clearly zero since it takes an infinite time to do a finite amount of work [1]. Therefore, Carnot efficiency, cannot be a comparison standard for real heat engines. For a given system, the amount of exergy destruction related to constraints is unavoidable [18]. The irreversibilities originating from finite-time and finite-size constraints are important in the real thermal system
Fig. 5. Effects of F on dimensionless total exergy destruction, internal exergy destruction, and external exergy destruction of the cycle.
ð39Þ
Fig. 6 depicts the effect of F on mean temperature of passing streams through high-temperature and low-temperature heat exchangers. The mean temperature difference increases between the hot and cold streams for the heat exchangers, while the difference reduces between the high and low temperature working fluids. Therefore, the exergy destruction in the heat exchangers and hence total external exergy destruction is due to the large temperature difference between the heat sources flows and corresponding working fluid. Then, in order to investigate the effect of variations in mean temperatures on the rate of exergy destruction, a second formula will be proposed for heat exergy new definition in terms of mean temperatures. To extract this formula, T 0 =T H is written as the product of the temperatures ratios Eq. (41). The second and fourth terms of this product are due to external irreversibilities. By eliminating these two terms from the product, the effect of unavoidable external irreversibility will be omitted from the definition of Exergy. Then, the new definition is achieved as Eq. (42). At these equations, T H ; T L ; T W1 ; and T W2 , are the high temperature flow, low temperature flow, high temperature working flow, and low temperature working flow mean temperatures, respectively. The thermodynamic average temperature of the heat exchangers flows, are obtained using following Eq. (43). [7].
T0 E_ F ¼ Q_ H : 1 TH
ð40Þ
T 0 T 0 T L T W2 T W1 T 0 T W2 ¼ ¼ T H T L T W2 T W1 T H T L T W1
ð41Þ
Fig. 6. Effect of F on mean temperature of streams passing through heat exchangers.
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T 0 T W2 E_ F;New ¼ Q_ H : 1 T L T W1
!! ð42Þ
Table 5 Fuel exergy rate and product exergy rate of the systems. System
R Te
T¼
HTHE
cP ðTÞdT he hi T ¼ R Ti e se si cP ðTÞ dT Ti T
ð43Þ
T0 T0 Q_ H 1 ¼ E_ F;1 E_ P;1 ¼ Q_ H 1 TH T W1 T T 0 0 ¼ Q_ H T W1 T H
E_ D;2 ¼ E_ F;2 E_ P;2 T0 T0 _ net Q_ L 1 W ¼ Q_ H 1 T W1 T W2 T0 T0 Q_ L 1 E_ D;3 ¼ E_ F;3 E_ P;3 ¼ Q_ L 1 T W2 TL T T 0 0 ¼ Q_ L T L T W2
ð44Þ
ð45Þ
ð46Þ
According to the Exergy definition which is the maximum amount of system useful power that is produced between initial state and the final dead state, and the maximum useful power (reversible power) is produced when the processes are carried out reversibly, the cycle exergy destruction rate must equal zero. Therefore, Eq. (45) leads to following equation for reversible power.
_ net;rev ¼ Q_ H 1 T 0 Q_ L 1 T 0 W T W1 T W2
Brayton Cycle LTHE
To prove this formula, the heat engine is modeled as three separate systems including high temperature heat exchanger, Brayton cycle, and low temperature heat exchanger, as shown in Fig. 7. Supposing that, the exergy transfers to or from the heat exchangers via heat transfer, rather than fluid flows. The rates of fuel and product exergies of the new modeling can be calculated as in Table 5. Now, According to Eq. (26), the exergy destruction rates of considering systems can be formulated as in Eqs. (44)–(46).
E_ D;1
Fuel Exergy rate Q_ H 1 T 0 TH Q_ H 1 T 0 T W1 Q_ L 1 T 0
ð47Þ
Product Exergy rate Q_ H 1 T 0 T W1 _ net Q_ L 1 T 0 þ W T W2 T0 _ QL 1
T W2
TL
In addition, reversible power for steady Brayton cycle can be achieved using first law of thermodynamics as Eq. (48).
_ net;rev ¼ Q_ H Q_ L W
ð48Þ
Now, subtracting Eq. (47) from Eq. (48), resulting in following equation.
Q_ H Q_ L ¼ T W1 T W2
ð49Þ
The Eq. (49), is the second law of thermodynamic for internal reversible cycle. Using Eqs. (44), (46), and (49), the total heat sources exergy destruction rate is obtained as Eq. (50).
E_ D;HeatSources ¼ E_ D;1 þ E_ D;2 ¼ Q_ H
T 0 T W2 T L T W1
!
T0 TH
!
ð50Þ
Therefore, according to Eqs. (38) and (50), the second formula for new defined heat exergy (Eq. (42).) is obtained for the second time, as Eq. (51).
E_ F;New ¼ E_ F E_ D;HeatSources ¼ Q_ H ¼ Q_ H 1 TT0 TT W2 L
1 TT 0 TT0 TT W2 TT 0 H
L
W1
H
W1
ð51Þ Based on the new defined thermal exergy, the second law efficiency definition is revised in Eq. (52).
_ W
gII ¼ _ Net EF;New
Fig. 7. The schematic of the three separated systems.
ð52Þ
M.M. Naserian et al. / Energy Conversion and Management 134 (2017) 116–124
The effect of F on dimensionless fuel exergy, dimensionless power, dimensionless total exergy destruction in thermal sources and the new definition of dimensionless thermal exergy is tracked in Fig. 8. These parameters are at their minimum value for F = 0.001. Fuel exergy and the total exergy destruction constantly increase with F. The maximum power initially increases with F and it decreases at higher values of F. For F smaller than 0.1, power and fuel exergy approximately take the same values and the difference between the two parameters increases afterwards due to the rapid growth of heat-transfer exergy destruction. At higher values of F, fuel exergy does not return an appropriate measure about attainable exergy due to neglecting the exergies of thermal sources (external irreversibility). The new definition presented for thermal exergy takes into account this deficiency. This model and fuel exergy follow the same pattern up to F = 0.1. The new exergy model decelerates thereafter and decreases after its maximum value at F = 0.5. The new exergy definition (Eq. (51)) is compared with the theoretical definition (subtracting exergy destruction of thermal sources from fuel exergy), this relationship is also drawn in the Figure as a function of F. The two definitions are consistent with each other to a great deal with small deviations at high values of F, which verifies the accuracy of calculations in current research. Fig. 9 shows variations of the first law efficiency, and internal reversible efficiency for maximum power state with F. The internal reversible efficiency is the first law efficiency of the heat engine when it is internally reversible. The first law efficiency decreases with a mild slope up to F = 0.1. After that, the decrement slope of the first law efficiency increases as F gets higher and at last, it tends to zero. The decrease in the first law efficiency is due to the decrease in power and increase in input heat at the same time. The reason for difference between the first law efficiency and IR (Internal Reversible) one is internal entropy generation of the cycle. If internal exergy destruction does not exist, internal entropy won’t be generated in the system and IR efficiency will superimpose on the first law efficiency. CA efficiency is a special state of IR efficiency (for F = 0.3), if the cycle is internally reversible and externally irreversible. In this analysis, CA efficiency represents less than 3 percent error compared with IR efficiency (Eq. (53)) at F = 0.3.
T T W1
gI:R ¼ 1 W2 ðF ! 0Þ ) gCA ¼ 1
ð53Þ qffiffiffiffi
ðF ¼ 0:3Þ ) gI:R ¼ 1
T L T H
¼1
546:24 813:05
qffiffiffiffiffiffiffiffiffiffi 413:33 892:27
¼ 0:319
¼ 0:328
123
Fig. 9. Effect of F on the first law efficiency, and internal reversible efficiency for maximum power state.
Fig. 10. Effect of F on the second law efficiency and the new defined second law efficiency.
Fig. 10 shows the changes in the second law efficiency and the new defined second law efficiency (Eq. (52)), with F. Both efficiencies decrease with increment of F, so that when F approaches to 1, the efficiencies move towards zero. The reductive behavior of the second law efficiency is due to simultaneous increase of input exergy and decrease of produced power (as a result of increase in exergy destruction). Furthermore, as shown in Fig. 10, the new efficiency definition is greater than the conventional one, because the value of the new exergy definition is smaller than the conventional one. For example at F = 0.3 the second law efficiency for the new and the prevalent definitions are obtained 0.528 and 0.365, respectively. Which means exergy overestimation in conventional definition, due to not considering the effect of unavoidable thermal sources exergy destructions on the conventional heat exergy definition, causes about 31% lower estimation of the second law efficiency ((0.528–0.365)/0.528 = 0.309). 4. Conclusion
Fig. 8. Effect of F on dimensionless Maximum power, Fuel Exergy, New defined exergy, and heat sources exergy destructions.
The optimal performance of a closed regenerative Brayton cycle was sought through power maximization. Optimization was performed on the output power as the objective function using genetic
124
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algorithm. The optimization was performed based on the dimensionless mass flow rate parameter (F). The behavior of the system variables, such as maximum power, and exergy. were investigated using F. The irreversibilities originating from finite-time and finitesize constraints are unavoidable in the real thermal system processes. In order to avoid these exergy destructions, heat exergy, and second law efficiency definitions were modified using two different ways. Moreover, the new definitions were compared with the conventional ones. For example, at a specified dimensionless mass-flow parameter, exergy overestimation in conventional definition, caused about 31% lower estimation of the second law efficiency. The results could be expected to be utilized in future solar thermal Brayton cycle studies.
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[18] Petrakopoulou F, Tsatsaronis G, Morosuk T, Carassai A. Conventional and advanced exergetic analyses applied to a combined cycle power plant. Energy 2012;41:146–52. [19] Vucˇkovic´ GD, Vukic´ MV, Stojiljkovic´ MM, Vucˇkovic´ DD. Avoidable and unavoidable exergy destruction and exergoeconomic evaluation of the thermal processes in a real industrial plant. Therm Sci 2012;16:433–46. [20] Açıkkalp E, Aras H, Hepbasli A. Advanced exergoeconomic analysis of a trigeneration system using a diesel-gas engine modeling and optimization of combined cycle power plant based on exergoeconomic and environmental analyses. Appl Therm Eng 2014;67:388–95. [21] Naserian MM, Farahat S, Sarhaddi F. Finite time exergy analysis and multiobjective ecological optimization of a regenerative Brayton cycle considering the impact of flow rate variations. Energy Convers Manage 2015;103:790–800. [22] Naserian MM, Farahat S, Sarhaddi F. Exergoeconomic multi objective optimization and sensitivity analysis of a regenerative Brayton cycle. Energy Convers Manage 2016;117:95–105. [23] Goodarzi M. Comparative energy analysis on a new regenerative Brayton cycle. Energy Convers Manage 2016;120:25–31. [24] Ho CK, Conboy T, Ortega J, Afrin S, Gray A, Christian JM et al. High-Temperature Receiver Designs for Supercritical CO2 Closed-Loop Brayton Cycles. In: ASME 2014 8th International Conference on Energy Sustainability collocated with the ASME 2014 12th International Conference on Fuel Cell Science, Engineering and Technology (pp. V001T02A003-V001T02A003). American Society of Mechanical Engineers; 2014. [25] Kusterer K, Braun R, Moritz N, Lin G, Bohn D. Helium Brayton Cycles With Solar Central Receivers: Thermodynamic and Design Considerations. In: ASME Turbo Expo 2012: Turbine Technical Conference and Exposition (pp. 271–279). American Society of Mechanical Engineers. [26] Persichilli M, Kacludis A, Zdankiewicz E, Held T. Supercritical CO2 Power Cycle Developments and Commercialization: Why sCO2 can Displace Steam Ste; 2012. [27] Chen L, Yang B, Sun F. Finite time exergoeconomic performance of a real intercooled regenerated gas turbine cogeneration plant. Part 1: model description and parametric analyses. Int J Low-Carbon Tech 2014;9:29–37. [28] Dyreby J, Klein S, Nellis G, Reindl D. Design considerations for supercritical carbon dioxide Brayton cycles with recompression. J Eng Gas Turbines Power 2014;136(10):101701. [29] Ahmadi MH, Ahmadi MA, Feidt M. Performance optimization of a solar-driven multi-step irreversible brayton cycle based on a multi-objective genetic algorithm. Oil & Gas Science and Technology-Revue d’IFP Energies nouvelles; 2014. [30] Barrett MJ, Reid BM. System Mass Variation and Entropy Generation in 100kWe Closed-Brayton-Cycle Space Power Systems; 2004. [31] Le Roux WG, Bello-Ochende T, Meyer JP. A review on the thermodynamic optimisation and modelling of the solar thermal Brayton cycle. Renew Sustain Energy Rev 2013;28(2013):677–90. [32] Singh R, Miller SA, Rowlands AS, Jacobs PA. Dynamic characteristics of a directheated supercritical carbon-dioxide Brayton cycle in a solar thermal power plant. Energy 2013;50(2013):194–204. [33] Durmayaz A, Sogut OS, Sahin B, Yavuz H. Optimization of thermal systems based on finite-time thermodynamics and thermoeconomics. Prog Energy Combust Sci 2004;30(2):175–217. [34] Özel G, Açıkkalp E, Savasß AF, Yamık H. Novel thermoenvironmental evaluation criteria and comparing them for an actual heat engine. Energy Convers Manage 2015;106:1118–23. [35] Açıkkalp E, Yamık H. Modeling and optimization of maximum available work for irreversible gas power cycles with temperature dependent specific heat. J Non-Equilib Thermodyn 2015;40(1):25–39. [36] Açikkalp E. Models for optimum thermo-ecological criteria of actual thermal cycles. Therm Sci 2013;17(3):915–30. [37] Açıkkalp E, Yamık H. Limits and optimization of power input or output of actual thermal cycles. Entropy 2013;15(8):3219–48. [38] Ahmadi P, Dincer I. Thermodynamic and exergoenvironmental analyses, and multi-objective optimization of a gas turbine power plant. Appl Therm Eng 2011;31:2529–40.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
New exergy analysis of a regenerative closed 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 10 August 2016 Received in revised form 14 November 2016 Accepted 10 December 2016
Keywords: Power Regenerative Brayton cycle Exergy Efficiency Optimization
a b s t r a c t In this study, the optimal performance of a regenerative closed Brayton cycle is sought through power maximization. Optimization is performed on the output power as the objective function using genetic algorithm. In order to take into account the time and the size constraints in current problem, the dimensionless mass-flow parameter is used. The influence of the unavoidable exergy destruction due to finitetime constraint is taken into account by developing the definition of heat exergy. Finally, the improved definitions are proposed for heat exergy, and the second law efficiency. Moreover, the new definitions will be compared with the conventional ones. For example, at a specified dimensionless mass-flow parameter, exergy overestimation in conventional definition, causes about 31% lower estimation of the second law efficiency. These results could be expected to be utilized in future solar thermal Brayton cycle assessment and optimization. Ó 2016 Elsevier Ltd. All rights reserved.
0. Introduction The underlying functional point of cyclic heat engines is maximum power state. The operational point in maximum power and reversible performance possess the same importance. The efficiency of heat engines are restricted by Carnot efficiency. This efficiency is obtainable in the reversible case. Practically, all thermodynamic processes take place in finite-size components during finite-time, which leads to irreversibility (exergy destruction). Accordingly, while Carnot cycle gives upper bound for thermal efficiency, it cannot be a comparison standard for real heat engines. Analysis techniques have been developed in various studies to consider the internal and/or external irreversibility in heat engines. Curzon and Ahlborn [1] studied the effect of external irreversibility, which accounted for irreversibilities in the heatexchange processes between the power cycle and its heat sources, on Carnot cycle’s output power and thermal efficiency. This system was entitled endoreversible due to internal reversibility of cycle. In their research, thermal efficiency at the maximum power state was expressed in the form of Eq. (1). T L and T H are temperatures at cold and hot heat exchangers, respectively.
gCA
sffiffiffiffiffiffi TL ¼1 TH
⇑ Corresponding author. E-mail address: [email protected] (S. Farahat). http://dx.doi.org/10.1016/j.enconman.2016.12.020 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.
ð1Þ
Bejan [2] showed that the degree of thermodynamic imperfection of power plants could be estimated based on a very simple model that considers only the sources of heat transfer irreversibilities. In a separate study, Bejan [3] investigated the optimal allocation of heat exchange equipment. His study showed that the power output of various power plant configurations could be maximized by properly dividing the fixed inventory of heat exchange equipment among the heat transfer components of each plant. Wu [4] established a comparison between endoreversible Carnot cycle and the same system with both internal and external irreversibility. It was shown that the internal irreversibility reduces power and efficiency. Gordon [5] analyzed heat engines considering finite rate heat transfer and finite-capacity thermal reservoirs. He showed that the efficiency at maximum power depends on the thermal reservoir temperatures, and other system variables such as reservoir capacity or working fluid specific heats. Heat engines operate in finite time; therefore, the realistic study of their optimal performance is feasible through the concept of finite-time thermodynamics [6]. This method was applied to the optimization of regenerative endoreversible Brayton cycle with finite thermal capacitance rates in heat reservoirs [8]. In the undertaken research, application of regenerators led to decrease in the maximum power and thermal efficiency. Their study showed the regenerative heat-transfer rate was positive for low temperature ratios and negative for high temperature ratios. Further analyses were performed on regenerative and irreversible models of Brayton heat engines [9,10].
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Nomenclature T _ W S_ r E_ e cP P E_ _ m Q_ A U F R PR
temperature (K) power (kW) entropy rate (kW/K) thermal conductance ratio 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/m 2K) dimensionless mass flow rate specific gas constant (J/kg K) pressure ratio
Greek symbols g efficiency e effectiveness
Optimization of real systems is confined to thermal performance and physical constraints. Bejan [11] established two optimization approaches based on these 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. [13] used heat exchanger size as design constraints for irreversible regenerative Brayton cycle optimization. In this model, finite-time thermodynamics and optimization were used to determine the maximum power and minimum entropy generation, along with the global maximum net power. In various research studies different thermal parameters were used for thermodynamic optimization of heat engines. Power density and exergy density are two of these objective functions, applied to irreversible Brayton cycle with regeneration, and cogeneration system [12,14]. Parametric design for the maximum exergy density leads to a smaller and more efficient cogeneration system. Yang et al. [15] showed that regeneration and intercooling process can improve the economic profit rate and exergy efficiency of cogeneration plants. Haesli [16] has studied second law efficiency optimization of a regenerative Brayton cycle. He examined the maximum limit of the thermal efficiency of an ideal Brayton cycle, and defined the second law efficiency of the cycle as the ratio of the thermal (first law) efficiency to the maximum attainable efficiency. Tsatsaronis and Park [17] have divided exergy destruction into avoidable and unavoidable parts and demonstrated how to estimate the avoidable and unavoidable exergy destruction associated with system’s components. F. Petrakopoulou et al. [18] analyzed a combined cycle power plant using both conventional and advanced exergetic analyses. They found that most of the exergy destruction in the plant components was unavoidable. Vucˇkovic´ et al. [19] investigated an industrial plant using both conventional and advanced exergy analysis. They claimed that highest exergy destruction was caused by the steam boiler. Moreover 92.34% of the total exergy destruction in boiler was unavoidable. Açıkkalp et al. [20] have analyzed a trigeneration system using an advanced exergy analysis. The results of their research indicated that the improvement potential of their system was low because 82% of the total exergy destruction cost rates were unavoidable. Naserian et al. sought the optimal performance of a regenerative Brayton
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 LTHE low temperature heat exchanger AC air compressor R heat regenerator HTHE high temperature heat exchanger
cycle through multi-objective ecological function maximization [21], and exergoeconomic multi objective optimization [22] using finite-time thermodynamic concept and finite-size components. Goodarzi [23] introduced and then analyzed energetically a new regenerative Brayton cycle. In current study, an irreversible closed-cycle Brayton engine with regeneration is investigated. Use of a closed-cycle system permits working fluids other than air, such as carbon dioxide [24], Helium [25]. Traditional Brayton cycles can be upgraded to supercritical CO2 to enable much greater efficiencies and power outputs [26]. Many studies have been published on the performance and optimization of the Brayton cycle and solar thermal Brayton cycle [27–32]. 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]. The main originality of the current research is as follows: The maximum attainable dimensionless net output power is obtained considering the impact of mass-flow parameter variation. The influence of dimensionless mass-flow parameter on internal, and external exergy destructions, and 1st and 2nd law efficiencies are investigated. The definitions of heat exergy, and second law efficiency are modified. 1. Heat engine model The schematic of the system, an irreversible regenerative closed Brayton cycle, is shown in Fig. 1. The heat engine includes an irreversible compressor, a regenerator, two heat exchangers, and an irreversible turbine. Heat exchangers are used to transfer heat from high temperature flow to the cycle and from the cycle to the low temperature flow. The entire analysis of system can be broken down into: (i) energy analysis (ii) exergy analysis. 1.1. Energy analysis The following assumptions are taken into account to develop thermodynamic model: All processes are steady state. Compressor
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Fig. 1. The schematic of the system.
and turbine are adiabatic. The efficiencies of the compressor and the turbine are considered 0.9 and 0.95, respectively. The temperature of working flow is variable. The chemical exergetic term does not vary. The kinetic and potential exergetic terms are neglected. The temperature of H1 and L1 flows are assumed 1200 K and 300 K, respectively. The dead-state condition is considered P0 ¼ 1:01 bar and T0 ¼ 293:15 K. The flowing gas is considered as a product of natural gas combustion process and it follows the principle of ideal-gas mixture. The heat capacity of the gas at constant pressure CP;gas varies with temperature as indicated in Eq. (2) [38].
cP;gas ðTÞ ¼ 0:93750 þ
0:01215 102
Tþ
0:01670 105
T2
0:07164 109
T3
_ m ðU:AÞT cP;min
i;0 < F < 1
ð3Þ
where cP;min is the least heat capacity of the gas in the heat engine. _ ¼ F:ððU:AÞT =cP;min Þ. Therefore, In the forthcoming equations m Energy balance equations for compressor (Eq. (4)), turbine (Eq. (5)), high temperature heat exchanger (Eq. (6)) and low temperature heat exchanger (Eq. (7)) are derived (in [21]) as follows.
8 <
39 cg 1 = 1 4 P 2 cg T2 ¼ T1 1 þ 15 : ; gc P1 8 <
_ P;LTHE min ðT 6 T L1 Þ ¼ eLTHE m:c
cP;HTHE min ¼ min cP;H ; cP;W1
ð8Þ
cP;LTHE min ¼ min cP;L ; cP;W2
ð9Þ
T H1 þ T 8 ¼ cP;gas 2
ð10Þ
cP;H
cP;L ¼ cP;gas
2
P4 T 5 ¼ T 4 1 þ ggt 41 : P5
39 1c cg = g 5 ;
ð5Þ
_ P;H ðT H1 T 8 Þ ¼ m:c _ P;W1 ðT 4 T 3 Þ Q_ H ¼ m:c _ P;HTHE min ðT H1 T 3 Þ ¼ eHTHE m:c
T L1 þ T 7 2
ð11Þ
cP;W1 ¼ cP;gas
T3 þ T4 2
ð12Þ
cP;W2 ¼ cP;gas
T6 þ T1 2
ð13Þ
Energy analysis of regenerator is studied in Eqs. (14)–(17).
_ P;R1 ðT 3 T 2 Þ ¼ mc _ P;R2 ðT 5 T 6 Þ Q_ R ¼ mc
eR ¼
T3 T2 T5 T2
ð14Þ ð15Þ
cP;R1 ¼ cP;g
T3 þ T2 2
ð16Þ
cP;R2 ¼ cP;g
T6 þ T1 2
ð17Þ
2
ð4Þ
ð7Þ
where the embedded parameters in above relations are described in Eqs. (8)–(13).
ð2Þ
In order to enter the size constraint to the analysis, total thermal conductance of system heat exchangers (ðU:AÞT ) is assumed to be constant [2,13,21,27,28]. Moreover, time will be constrained if the mass flow rate is finite and non-zero. For taking into account the time and the size constraints in current investigation, the dimensionless mass-flow parameter (F) is used. This parameter was defined by Naserian et al. [21], as Eq. (3).
F¼h
_ P;L ðT 7 T L1 Þ ¼ m:c _ P;W2 ðT 6 T 1 Þ Q_ L ¼ m:c
Finally, the net power in the heat engine is presented in Eqs. (18) and (19).
_ net ¼ Q_ H Q_ L W
ð18Þ
_ net ¼ eHTHE :m:c _ P;HTHE min :ðT H1 T 3 Þ eLTHE :m:c _ P;LTHE min :ðT 6 T L1 Þ W ð6Þ
ð19Þ
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1.2. Exergy analysis of system Exergy analysis helps to investigate the efficient part of energy at thermodynamic processes and the results of such analysis lead to operational improvements. In order to carry out an exergy analysis, an exergy model must be provided. Physical exergy for temperature dependent specific heat is defined as Eq. (20).
Z _ E_ ¼ m
Z
T
T
cP ðTÞdT T 0
T0
cP ðTÞ T0
dT P R ln T P0
ð20Þ
Physical exergy of flows (based on Eq. (2)) is expressed in Eqs. (21)–(25). For i ¼ H1; L1; 1; 2; 3; 4; 5; 6; 7; 8 02
_B E_ i ¼ m @4
1 3T i T 2 þ 0:00567 T 3 0:01791 T4 c0:93750T þ 0:006075 P 105 102 109 i 5 þ T 0 Rln C A 0:00835 2 0:02388 3 P0 T 0 0:93750lnðTÞ þ 0:01215 T þ T T 2 5 9 10 10 10 T0
ð21Þ
_ P;AC ðT 2 T 1 Þ E_ 9 ¼ mc
ð22Þ
_ P;gt ðT 4 T 5 Þ E_ 10 ¼ mc
ð23Þ
T4 þ T5 2
ð24Þ
T1 þ T2 ¼ cP;g 2
ð25Þ
cP;gt ¼ cP;g
cP;AC
1.2.1. Exergy of cycle components The fuel exergy rate (physical exergy related to the streams entering each component) and product exergy rate (physical exergy related to the streams exiting each component) for system components are tabulated in Table 1. Exergy destructions of system components are formulated in Eqs. (26)–(31) [7].
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, exergy, first law, and second law efficiencies corresponding to F, are going to be plotted. Finally, regarding the results, new definition of exergy, and second law efficiency will be obtained. 2. Optimization study For certain values of F, the system is simulated in MATLAB.. Outcome of this modeling is used in genetic algorithm for optimization purpose. Genetic algorithm has been used by several researchers (such as Naserian et al. [21,22], Ahmadi et al. [25], and Ahmadi et al. [29]) to optimize the operation of thermodynamic cycles. The flowchart of optimization process is shown in Fig. 2. The net output power of system is specified as the objective function for optimization (Eq. (33)).
_ ¼ E_ 10 E_ 9 W
ð33Þ
Maximize Eq. (33) The decision variables for optimization of the system are the compressor pressure ratio, and thermal conductance ratios of heat exchangers (Eqs. (34) and (35).
rHTHE ¼
ðU:AÞHTHE ðU:AÞT
ð34Þ
rLTHE ¼
ðU:AÞLTHE ðU:AÞT
ð35Þ
2.1. Constraints The significant constraint taken into account in this study is the heat exchangers’ size. The total thermal conductance of heat exchangers is constant, which results in Eq. (35), where r R is described as Eq. (37). The decision variables bounds, and the stopping criteria used for optimization problem are shown in Tables 3 and 4, respectively.
E_ D;k ¼ E_ F;k E_ P;k
ð26Þ
E_ D;1 ¼ E_ F;1 E_ P;1 ¼ E_ 9 ðE_ 2 E_ 1 Þ
ð27Þ
E_ D;2 ¼ E_ F;2 E_ P;2 ¼ ðE_ 5 E_ 6 Þ ðE_ 3 E_ 2 Þ
ð28Þ
E_ D;3 ¼ E_ F;3 E_ P;3 ¼ ðE_ H1 E_ 8 Þ ðE_ 4 E_ 3 Þ
ð29Þ
rR ¼
E_ D;4 ¼ E_ F;4 E_ P;4 ¼ ðE_ 4 E_ 5 Þ E_ 10
ð30Þ
The tuning parameters values for Genetic algorithm optimization are given in Table 2.
E_ D;5 ¼ E_ F;5 E_ P;5 ¼ ðE_ 6 E_ 1 Þ ðE_ 7 E_ L1 Þ
ð31Þ
ðU:AÞTOT ¼ Constant
ð36Þ
rHTHE þ rLTHE þ r R ¼ 1 0 < r < 1
ð37Þ
ðU:AÞR ðU:AÞT
ð38Þ
3. Results and discussion
The total exergy destruction of cycle is calculated in Eq. (32).
E_ D;T ¼
5 X E_ D;k
ð32Þ
k¼1
Table 1 Fuel and product exergy rates 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
Variation of dimensionless net power with parameter F is depicted in Fig. 3 at the maximum power state. Hereafter, all results are investigated at the maximum state even though it is not stated. The slope of power increases gradually with F up to the maximum of maximum power value at F = 0.3. Afterwards, the dimensionless net power slopes down sharply. Therefore, the maximum of maximum net power is located between the two limits. Exergy of high temperature thermal source, dimensionless input heat to the system ðQ_ H =ððU:AÞ T0 ÞÞ, dimensionless fuel exergy T
(input thermal exergy to the system) and dimensionless exergy of input heat to the system (exergy of fuel heat) are studied in Fig. 4 as a function of F. These parameters proportionally increase with the increase in F. Hot line exergy partially enters the system through high temperature heat exchanger and the rest exists within
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M.M. Naserian et al. / Energy Conversion and Management 134 (2017) 116–124 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 3 The bounds used for decision variables. Decision variable
Range
Pressure ratio of compressor r HTHE rLTHE
[1,20] [0,1] [0,1]
Table 4 Stop criteria for the optimization algorithms. Stop criterion
Value
Number of generation Function tolerance
1000 1e7
Fig. 3. Effect of F on the value of dimensionless maximum net power.
dimensionless fuel exergy on dimensionless exergy of fuel heat verifies the accuracy of energy and exergy calculations in current work.
3.1. Modification of heat exergy, and second law efficiency definitions
Fig. 2. The flowchart of optimization process.
fluid flow (the main objective of this study is to investigate input and output exergies of the cycle and their dependency on F; therefore, methods of using flow output exergy is not detailed here). Dimensionless of Q_ H , fuel exergy (E_ H1 E_ 8 ) and exergy of fuel heat (Q_ H ð1 T0 =TH Þ) behave similarly. The exact superimposition of
A system delivers the maximum possible work as it undergoes a reversible process from the specified initial state to the dead state. This represents the useful work potential of the system at the specified state and is called exergy. It is important to realize that exergy does not represent the amount of work that a work-producing device will actually deliver. There will always be a difference between exergy and the actual work delivered by a device. Irreversibilities such as friction, heat transfer through a finite temperature difference, nonquasiequilibrium compression or expansion and etc. always generate entropy, and anything that generates entropy always causes (the difference) exergy destruction [24]. Fig. 5 shows the changes in dimensionless total exergy destruction, internal exergy destruction ðE_ D;1 þ E_ D;2 þ E_ D;4 Þ, and external exergy
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121
optimization [33]. Size and time are the sources of entropy generation in practical situations [21]. One of the most important disadvantages of conventional definition of exergy is that it is overestimate the power of actual systems. Therefore, the conventional definition of the second law efficiency of the system is lower estimated. The unavoidable part of exergy destruction (even if with technology improvement), never can be overcome. In order to avoid these exergy destructions, the reasonable approach is to express exergy as a function of size and time. The influence of exergy destruction due to finite-time constraint is taken into account by eliminating the exergy destructions of thermal sources, which are unavoidable, from conventional exergy definition as Eq. (38).
E_ F;New ¼ E_ F E_ D;HeatSources
Fig. 4. Effect of F on dimensionless Hot line exergy, Fuel exergy, Exergy of fuel heat, and dimensionless input heat to the system.
destruction ðE_ D;3 þ E_ D;5 Þ of the cycle, relating to F. According to what is observed, all of the parameters are equal to zero for the lowest value of F, as the external exergy destruction and internal exergy destruction are equal to zero. Hence, the efficiency approximates the Carnot efficiency as the process time approaches infinity. The parameters increase with increment of F. Comparison of the diagrams proves that a great part of total exergy destruction is derived from external exergy destruction of the system (i.e. the heat exchangers of thermal sources). Nevertheless, due to significant increment of exergy destruction in thermal sources, exergy destruction increases progressively and it reaches the maximum value at F = 1. Carnot cycle gives upper bound for thermal efficiency. However, to achieve this theoretical efficiency, the heat transfer process have to be carried out infinitely slowly (in infinite time) or in infinitely large reservoirs (in infinite size) so that the working substance can come into thermal equilibrium with the heat sources. Under these conditions the power output is clearly zero since it takes an infinite time to do a finite amount of work [1]. Therefore, Carnot efficiency, cannot be a comparison standard for real heat engines. For a given system, the amount of exergy destruction related to constraints is unavoidable [18]. The irreversibilities originating from finite-time and finite-size constraints are important in the real thermal system
Fig. 5. Effects of F on dimensionless total exergy destruction, internal exergy destruction, and external exergy destruction of the cycle.
ð39Þ
Fig. 6 depicts the effect of F on mean temperature of passing streams through high-temperature and low-temperature heat exchangers. The mean temperature difference increases between the hot and cold streams for the heat exchangers, while the difference reduces between the high and low temperature working fluids. Therefore, the exergy destruction in the heat exchangers and hence total external exergy destruction is due to the large temperature difference between the heat sources flows and corresponding working fluid. Then, in order to investigate the effect of variations in mean temperatures on the rate of exergy destruction, a second formula will be proposed for heat exergy new definition in terms of mean temperatures. To extract this formula, T 0 =T H is written as the product of the temperatures ratios Eq. (41). The second and fourth terms of this product are due to external irreversibilities. By eliminating these two terms from the product, the effect of unavoidable external irreversibility will be omitted from the definition of Exergy. Then, the new definition is achieved as Eq. (42). At these equations, T H ; T L ; T W1 ; and T W2 , are the high temperature flow, low temperature flow, high temperature working flow, and low temperature working flow mean temperatures, respectively. The thermodynamic average temperature of the heat exchangers flows, are obtained using following Eq. (43). [7].
T0 E_ F ¼ Q_ H : 1 TH
ð40Þ
T 0 T 0 T L T W2 T W1 T 0 T W2 ¼ ¼ T H T L T W2 T W1 T H T L T W1
ð41Þ
Fig. 6. Effect of F on mean temperature of streams passing through heat exchangers.
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T 0 T W2 E_ F;New ¼ Q_ H : 1 T L T W1
!! ð42Þ
Table 5 Fuel exergy rate and product exergy rate of the systems. System
R Te
T¼
HTHE
cP ðTÞdT he hi T ¼ R Ti e se si cP ðTÞ dT Ti T
ð43Þ
T0 T0 Q_ H 1 ¼ E_ F;1 E_ P;1 ¼ Q_ H 1 TH T W1 T T 0 0 ¼ Q_ H T W1 T H
E_ D;2 ¼ E_ F;2 E_ P;2 T0 T0 _ net Q_ L 1 W ¼ Q_ H 1 T W1 T W2 T0 T0 Q_ L 1 E_ D;3 ¼ E_ F;3 E_ P;3 ¼ Q_ L 1 T W2 TL T T 0 0 ¼ Q_ L T L T W2
ð44Þ
ð45Þ
ð46Þ
According to the Exergy definition which is the maximum amount of system useful power that is produced between initial state and the final dead state, and the maximum useful power (reversible power) is produced when the processes are carried out reversibly, the cycle exergy destruction rate must equal zero. Therefore, Eq. (45) leads to following equation for reversible power.
_ net;rev ¼ Q_ H 1 T 0 Q_ L 1 T 0 W T W1 T W2
Brayton Cycle LTHE
To prove this formula, the heat engine is modeled as three separate systems including high temperature heat exchanger, Brayton cycle, and low temperature heat exchanger, as shown in Fig. 7. Supposing that, the exergy transfers to or from the heat exchangers via heat transfer, rather than fluid flows. The rates of fuel and product exergies of the new modeling can be calculated as in Table 5. Now, According to Eq. (26), the exergy destruction rates of considering systems can be formulated as in Eqs. (44)–(46).
E_ D;1
Fuel Exergy rate Q_ H 1 T 0 TH Q_ H 1 T 0 T W1 Q_ L 1 T 0
ð47Þ
Product Exergy rate Q_ H 1 T 0 T W1 _ net Q_ L 1 T 0 þ W T W2 T0 _ QL 1
T W2
TL
In addition, reversible power for steady Brayton cycle can be achieved using first law of thermodynamics as Eq. (48).
_ net;rev ¼ Q_ H Q_ L W
ð48Þ
Now, subtracting Eq. (47) from Eq. (48), resulting in following equation.
Q_ H Q_ L ¼ T W1 T W2
ð49Þ
The Eq. (49), is the second law of thermodynamic for internal reversible cycle. Using Eqs. (44), (46), and (49), the total heat sources exergy destruction rate is obtained as Eq. (50).
E_ D;HeatSources ¼ E_ D;1 þ E_ D;2 ¼ Q_ H
T 0 T W2 T L T W1
!
T0 TH
!
ð50Þ
Therefore, according to Eqs. (38) and (50), the second formula for new defined heat exergy (Eq. (42).) is obtained for the second time, as Eq. (51).
E_ F;New ¼ E_ F E_ D;HeatSources ¼ Q_ H ¼ Q_ H 1 TT0 TT W2 L
1 TT 0 TT0 TT W2 TT 0 H
L
W1
H
W1
ð51Þ Based on the new defined thermal exergy, the second law efficiency definition is revised in Eq. (52).
_ W
gII ¼ _ Net EF;New
Fig. 7. The schematic of the three separated systems.
ð52Þ
M.M. Naserian et al. / Energy Conversion and Management 134 (2017) 116–124
The effect of F on dimensionless fuel exergy, dimensionless power, dimensionless total exergy destruction in thermal sources and the new definition of dimensionless thermal exergy is tracked in Fig. 8. These parameters are at their minimum value for F = 0.001. Fuel exergy and the total exergy destruction constantly increase with F. The maximum power initially increases with F and it decreases at higher values of F. For F smaller than 0.1, power and fuel exergy approximately take the same values and the difference between the two parameters increases afterwards due to the rapid growth of heat-transfer exergy destruction. At higher values of F, fuel exergy does not return an appropriate measure about attainable exergy due to neglecting the exergies of thermal sources (external irreversibility). The new definition presented for thermal exergy takes into account this deficiency. This model and fuel exergy follow the same pattern up to F = 0.1. The new exergy model decelerates thereafter and decreases after its maximum value at F = 0.5. The new exergy definition (Eq. (51)) is compared with the theoretical definition (subtracting exergy destruction of thermal sources from fuel exergy), this relationship is also drawn in the Figure as a function of F. The two definitions are consistent with each other to a great deal with small deviations at high values of F, which verifies the accuracy of calculations in current research. Fig. 9 shows variations of the first law efficiency, and internal reversible efficiency for maximum power state with F. The internal reversible efficiency is the first law efficiency of the heat engine when it is internally reversible. The first law efficiency decreases with a mild slope up to F = 0.1. After that, the decrement slope of the first law efficiency increases as F gets higher and at last, it tends to zero. The decrease in the first law efficiency is due to the decrease in power and increase in input heat at the same time. The reason for difference between the first law efficiency and IR (Internal Reversible) one is internal entropy generation of the cycle. If internal exergy destruction does not exist, internal entropy won’t be generated in the system and IR efficiency will superimpose on the first law efficiency. CA efficiency is a special state of IR efficiency (for F = 0.3), if the cycle is internally reversible and externally irreversible. In this analysis, CA efficiency represents less than 3 percent error compared with IR efficiency (Eq. (53)) at F = 0.3.
T T W1
gI:R ¼ 1 W2 ðF ! 0Þ ) gCA ¼ 1
ð53Þ qffiffiffiffi
ðF ¼ 0:3Þ ) gI:R ¼ 1
T L T H
¼1
546:24 813:05
qffiffiffiffiffiffiffiffiffiffi 413:33 892:27
¼ 0:319
¼ 0:328
123
Fig. 9. Effect of F on the first law efficiency, and internal reversible efficiency for maximum power state.
Fig. 10. Effect of F on the second law efficiency and the new defined second law efficiency.
Fig. 10 shows the changes in the second law efficiency and the new defined second law efficiency (Eq. (52)), with F. Both efficiencies decrease with increment of F, so that when F approaches to 1, the efficiencies move towards zero. The reductive behavior of the second law efficiency is due to simultaneous increase of input exergy and decrease of produced power (as a result of increase in exergy destruction). Furthermore, as shown in Fig. 10, the new efficiency definition is greater than the conventional one, because the value of the new exergy definition is smaller than the conventional one. For example at F = 0.3 the second law efficiency for the new and the prevalent definitions are obtained 0.528 and 0.365, respectively. Which means exergy overestimation in conventional definition, due to not considering the effect of unavoidable thermal sources exergy destructions on the conventional heat exergy definition, causes about 31% lower estimation of the second law efficiency ((0.528–0.365)/0.528 = 0.309). 4. Conclusion
Fig. 8. Effect of F on dimensionless Maximum power, Fuel Exergy, New defined exergy, and heat sources exergy destructions.
The optimal performance of a closed regenerative Brayton cycle was sought through power maximization. Optimization was performed on the output power as the objective function using genetic
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algorithm. The optimization was performed based on the dimensionless mass flow rate parameter (F). The behavior of the system variables, such as maximum power, and exergy. were investigated using F. The irreversibilities originating from finite-time and finitesize constraints are unavoidable in the real thermal system processes. In order to avoid these exergy destructions, heat exergy, and second law efficiency definitions were modified using two different ways. Moreover, the new definitions were compared with the conventional ones. For example, at a specified dimensionless mass-flow parameter, exergy overestimation in conventional definition, caused about 31% lower estimation of the second law efficiency. The results could be expected to be utilized in future solar thermal Brayton cycle studies.
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