Open AccessArticle

Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle

Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China

School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China

College of Power Engineering, Naval University of Engineering, Wuhan 430033, China

Authors to whom correspondence should be addressed.

Entropy 2021, 23(3), 282; https://doi.org/10.3390/e23030282

Submission received: 17 January 2021 / Revised: 18 February 2021 / Accepted: 22 February 2021 / Published: 26 February 2021

(This article belongs to the Special Issue Carnot Cycle and Heat Engine Fundamentals and Applications II)

Download

Browse Figures

Versions Notes

Abstract

An improved irreversible closed modified simple Brayton cycle model with one isothermal heating process is established in this paper by using finite time thermodynamics. The heat reservoirs are variable-temperature ones. The irreversible losses in the compressor, turbine, and heat exchangers are considered. Firstly, the cycle performance is optimized by taking four performance indicators, including the dimensionless power output, thermal efficiency, dimensionless power density, and dimensionless ecological function, as the optimization objectives. The impacts of the irreversible losses on the optimization results are analyzed. The results indicate that four objective functions increase as the compressor and turbine efficiencies increase. The influences of the latter efficiency on the cycle performances are more significant than those of the former efficiency. Then, the NSGA-II algorithm is applied for multi-objective optimization, and three different decision methods are used to select the optimal solution from the Pareto frontier. The results show that the dimensionless power density and dimensionless ecological function compromise dimensionless power output and thermal efficiency. The corresponding deviation index of the Shannon Entropy method is equal to the corresponding deviation index of the maximum ecological function.

Keywords:

closed simple Brayton cycle; power output; thermal efficiency; power density; ecological function; multi-objective optimization

1. Introduction

Some scholars have studied performances of gas turbine plants (Brayton cycle (BCY)) [1,2,3,4] all over the world for their small size and comprehensive energy sources. The gas-steam combined, cogeneration, and other complex cycles have appeared for the requirements of energy conservation and environmental protection. The thermal efficiency (

η

) of a simple BCY is low, and the NOx content in combustion product is high. To further improve the cycle performance, it has become a key research direction to improve the initial temperature of the gas or to adopt the advanced cycles (such as regenerative, intercooled, intercooled and regenerative, isothermal heating, and other complex combined cycles).

In the case of simple heating, when the compressible subsonic gas flows through the smooth heating pipe with the fixed cross-sectional area, the gas temperature increases along the pipe direction; in the case of simple region change, when the compressible subsonic gas flows through the smooth adiabatic reductive pipe, the gas temperature decreases along the pipe direction. Based on these two gas properties, the isothermal heating process (IHP) can be realized when the compressible subsonic gas flows through the smooth heating reductive pipe. The combustion chamber, which can recognize the IHP, is called the convergent combustion chamber (CCC). The pipe of the CCC is assumed to be smooth. During the heating process, the temperature of the gas is always constant. According to the energy conservation law, the kinetic energy of the gas increases, that is, the pushing work of the gas increases. From the definition of enthalpy, it can be seen that enthalpy includes two parts: the thermodynamic energy and the pushing work. Therefore, the enthalpy increases. Based on this, Vecchiarelli et al. [5] proposed the CCC to perform the IHP of the working fluid. The power output (

\bar{W}

) and

η

of the BCY could be improved, and the emission of harmful gases such as NOx could be reduced by adding this combustion chamber model. The regenerative BCYs [6,7,8] and binary BCY [9] with IHPs were also studied by applying the classical thermodynamics.

Finite time thermodynamics (FTT) is a useful thermodynamic analysis theory and method [10,11,12,13,14,15,16,17,18,19]. In general, it is known that Curzon and Ahlborn [12] initialized FTT in 1975. In fact, the classical efficiency bound at the maximum power was also derived by Moutier [10] in 1872 and Novikov [11] in 1957. The applications of FTT include majorly two fields: optimal configurations [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] and optimal performances [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] studies for thermodynamic cycles and processes. The

W

and

η

have been often considered as the optimization objectives (OPOs) of the heat engines [62,63,64,65,66,67,68,69,70,71,72]. When the power density (

P

) [73,74,75,76,77,78,79,80,81] was taken as the OPO, the operating unit had a smaller size and higher

η

. Aditionally, the ecological function (

E

) [82,83,84,85,86,87,88] is also an OPO that balances the conflict between

W

and

η

Kaushik et al. [89] first applied the FTT to studying the regenerative BCY with an IHP. The regenerative, intercooled and regenerative complex BCYs with isothermal heating combustor were further investigated [90,91,92,93,94,95,96]. Based on this, Chen et al. [97,98,99] studied the endoreversible simple isothermal heating BCY with the

W

η

and

E

as OPOs. Arora et al. [100,101] adopted NSGA-II and evolutionary algorithms to optimize the irreversible isothermal heating regenerative BCY with the

W

and

η

as the OPOs. Chen et al. [102] considered the variable isothermal pressure drop ratio (

π_{t}

), established an improved isothermal heating regenerative BCY model, and studied the regenerator’s role on cycle performance. Qi et al. [103] demonstrated a closed endoreversible modified binary BCY with IHPs and found the

W

and

η

raised as the heat reservoirs’ temperature ratios. Tang et al. [104] considered the variable

π_{t}

and established an improved irreversible binary BCY model modified by isothermal heating. The heat exchanger’s heat conductance distributions (HCDs) and the top and bottom cycles’ pressure ratios were taken as optimization variables to optimize the cycle performance.

In the process of the thermodynamic system optimization, single-objective optimization often led to unacceptable objectives for other objectives when there were conflicts among the considered goals. Multi-objective optimization would consider the trade-offs among the goals, and the optimized results were more reasonable [99,100,102,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125].

In applying the FTT, the heat transfer was introduced into the thermodynamic analysis of the thermodynamic process, and finite temperature difference was considered in Refs. [11,12]. In this paper, the same method in Refs. [11,12] will be used, and the finite temperature difference will be considered when establishing the model, which is the key relation among this paper and the Refs. [11,12]. On this basis, the cycle’s irreversibility will be further considered, and the corresponding conclusion will be more in line with the actual situation. The compression and expansion losses in the model in Refs. [97,98,99] were not considered, and they will be further considered in this paper alongside the losses in the heat exchangers. Meanwhile, the thermal resistance loss and the optimal HCD will be considered. With the

W

η

P

and

E

, respectively, as the OPOs, an improved irreversible closed modified simple BCY with one IHP and coupled to variable-temperature heat reservoirs (VTHRs) will be optimized, and the optimization results will be compared. The effects of the compressor and turbine efficiencies on optimization results will be analyzed. The NSGA-II algorithm will be applied for multi-objective optimization to obtain the Pareto frontier further. The results obtained in this paper will reveal the original results in Refs. [10,11,12], which were the initial work of the FTT theory.

2. Cycle Model and Performance Analytical Indicators

The schematic diagram of an improved irreversible closed modified simple BCY with one IHP and coupled to VTHRs is shown in Figure 1. A compressor (C), a regular combustion chamber (RCC), a CCC, a turbine (T), and a precooler are the main parts of the cycle. The corresponding

T - s

diagram of the cycle is shown in Figure 2. The cycle consists of five processes in total:

The process $1 \to 2$ is an irreversible adiabatic compression process in C, and the process $1 \to 2 s$ is an isentropic process corresponding to the process $1 \to 2$ .
The process $2 \to 3$ is an isobaric endothermic process in RCC.
The process $3 \to 4$ is an IHP in CCC. In CCC, the working fluid is isothermally heated, and its flow velocity rises from $V_{3}$ to $V_{4}$ (the Mach number increases from $M_{3}$ to $M_{4}$ ), and its specific enthalpy rises from $h_{3}$ to $h_{4}$ . The parameter $π_{t} (= p_{4} / p_{3} \leq 1)$ is the isothermal pressure drop ratio. The $π_{t}$ needs to be given in Refs. [97,98], but the $π_{t}$ of the improved cycle established in this paper will change with the operation state. The degree of the IHP can be represented by $π_{t}$ , and the greater the $π_{t}$ , the greater the degree.
The process $4 \to 5$ is an adiabatic exothermic process in turbine, and the process $4 \to 5 s$ is the isentropic process corresponding to the process $4 \to 5$ .
The process $5 \to 1$ is an isobaric exothermic process in a precooler.

The working fluid is the ideal gas. The pressures and temperatures of the working fluid are

p_{i} (i = 1, 2, 3, 4, 5, 2 s, 5 s)

and

T_{i}

, and the ratio of specific heat is

k

. The outside fluids’ temperatures are

T_{j} (j = H 1, H 2, H 3, H 4, L 1, L 2)

. The specific heat at constant pressure and the working fluid’s mass flow rate are

C_{p}

and

\dot{m}

. The working fluid’s thermal capacity rate is

C_{w f}

where

C_{w f} = C_{p} \dot{m}

. The outer fluids’ thermal capacity rates at the RCC, CCC, and precooler are

C_{H}

C_{H 1}

and

C_{L}

, respectively; then, one has:

C_{H \max} = \max {C_{H}, C_{w f}}, C_{L \max} = \max {C_{L}, C_{w f}}, C_{H \min} = \min {C_{H}, C_{w f}}, C_{L \min} = \min {C_{L}, C_{w f}}

(1)

The heat exchangers’ heat conductance is the product of the heat transfer coefficient and the heat transfer area. The heat exchangers’ heat conductance values in the RCC, CCC, and precooler are

U_{H}

U_{H 1}

and

U_{L}

, the heat transfer units’ numbers are

N_{H}

N_{H 1}

and

N_{L}

, and the effectiveness values are

E_{H}

E_{H 1}

and

E_{L}

, respectively:

N_{H} = U_{H} / C_{H \min}, N_{H 1} = U_{H 1} / C_{H 1}, N_{L} = U_{L} / C_{L \min}

(2)

E_{H} = \frac{1 - e^{- N_{H} (1 - C_{H \min} / C_{H \max})}}{1 - (C_{H \min} / C_{H \max}) e^{- N_{H} (1 - C_{H \min} / C_{H \max})}}

(3)

E_{H 1} = 1 - e^{- N_{H 1}}

(4)

E_{L} = \frac{1 - e^{- N_{L} (1 - C_{L \min} / C_{L \max})}}{1 - (C_{L \min} / C_{L \max}) e^{- N_{L} (1 - C_{L \min} / C_{L \max})}}

(5)

When

C_{H \max} = C_{H \min}

and

C_{L \max} = C_{L \min}

, Equations (3) and (5) are, respectively, simplified as:

E_{H} = N_{H} / (N_{H} + 1)

(6)

E_{L} = N_{L} / (N_{L} + 1)

(7)

The outside fluids’ temperature ratios at the RCC and CCC are:

τ_{H 1} = T_{H 1} / T_{0}

(8)

τ_{H 3} = T_{H 3} / T_{0}

(9)

where

T_{0}

is the ambient temperature.

The process

1 \to 2 s

is the isentropic one, namely:

T_{2 s} / T_{1} = π^{m} = x

(10)

where

m = (k - 1) / k

and

π

is the pressure ratio of the compressor.

The process

4 \to 5 s

is the isentropic one, namely:

T_{4} / T_{5 s} = π^{m} π_{t}^{m} = x y

(11)

The process

3 \to 4

is the isothermal one, namely:

T_{3} = T_{4}

(12)

{\dot{Q}}_{3 - 4} = \dot{m} (h_{4} - h_{3}) - \dot{m} \int_{3}^{4} v d p = - \dot{m} R_{g} T_{3} \ln π_{t}

(13)

where

π_{t}

M_{3}

and

M_{4}

must satisfy the following relation:

\ln π_{t} = - c_{p} (k - 1) (M_{4}^{2} - M_{3}^{2}) / (2 R_{g})

(14)

where the working fluid’s flow velocity must be subsonic, namely,

M_{3}, M_{4} < 1

Because the working fluid has an initial speed,

(M_{4}^{2} - M_{3}^{2}) < 0.96

and

π_{t} > 0.5107

when

M_{3} = 0.2

. Because of

M_{4} > M_{3}

π_{t} < 1

. When

π_{t} = 1

, the cycle model in this paper can be simplified to a simple Brayton cycle.

According to the definition of

π_{t}

, it can be obtained that:

π_{t} = \frac{p_{4}}{p_{3}} = \frac{p_{4}}{p_{3}} \cdot \frac{p_{1}}{p_{1}} = \frac{p_{4}}{p_{1}} \cdot π^{- 1} \geq π^{- 1}

(15)

Considering the irreversibilities in the compressor and the turbine, the efficiencies of them are:

η_{c} = (T_{1} - T_{2 s}) / (T_{1} - T_{2})

(16)

η_{t} = (T_{5} - T_{4}) / (T_{5 s} - T_{4})

(17)

The pressure drop is not considered in this paper. It will be considered in future, as it was by Ref. [126]. The study in Ref. [126] showed that the pressure drop loss has a little influence on the cycle performance quantitatively, and has no influence qualitatively.

The working fluid’s heat absorption rates at RCC and CCC are

{\dot{Q}}_{2 - 3}

and

{\dot{Q}}_{3 - 4}

, respectively:

{\dot{Q}}_{2 - 3} = C_{H} (T_{H 1} - T_{H 2}) = C_{w f} (T_{3} - T_{2}) = C_{H \min} E_{H} (T_{H 1} - T_{2})

(18)

{\dot{Q}}_{3 - 4} = C_{H 1} (T_{H 3} - T_{H 4}) = C_{H 1} E_{H 1} (T_{H 3} - T_{3}) = \dot{m} (V_{^{4}}^{2} - V_{3}^{2}) / 2

(19)

The heat releasing rate at the precooler is

{\dot{Q}}_{5 - 1}

, namely:

{\dot{Q}}_{5 - 1} = C_{L} (T_{L 2} - T_{L 1}) = C_{w f} (T_{5} - T_{1}) = C_{L \min} E_{L} (T_{5} - T_{L 1})

(20)

The heat leakages between the heat source and the environment [127,128] are neglected. Therefore, the

W

and

η

are:

W = {\dot{Q}}_{2 - 3} + {\dot{Q}}_{3 - 4} - {\dot{Q}}_{5 - 1}

(21)

η = W / ({\dot{Q}}_{2 - 3} + {\dot{Q}}_{3 - 4})

(22)

The dimensionless power output (

\bar{W}

) is:

\bar{W} = W / (C_{w f} T_{0})

(23)

The maximum specific volume corresponding to state point 5 is

v_{5}

. The

P

is calculated as:

P = W / v_{5}

(24)

The specific volume corresponding to state point 1 is

v_{1}

. The dimensionless power density (

\bar{P}

) and dimensionless maximum specific volume (

v_{5} / v_{1}

) are obtained as:

\bar{P} = \frac{P}{C_{w f} T_{0} / v_{1}} = \frac{W / v_{5}}{C_{w f} T_{0} / v_{1}} = \frac{W}{C_{w f} T_{0}} \times \frac{T_{1}}{T_{5}} = \bar{W} \times \frac{T_{1}}{T_{5}}

(25)

v_{5} / v_{1} = T_{5} / T_{1}

(26)

There are two different methods for calculating the entropy production rate. One was suggested by Bejan [129,130], and the another was suggested by Salamon et al. [131]. In this article, the method used is the one suggested by the latter.

The entropy production rate (

s_{g}

) and

E

are, respectively, calculated as:

s_{g} = C_{H} \ln (T_{H 2} / T_{H 1}) + C_{H 1} \ln (T_{H 4} / T_{H 3}) + C_{L} \ln (T_{L 2} / T_{L 1})

(27)

E = W - T_{0} s_{g}

(28)

The dimensionless ecological function (

\bar{E}

) is obtained as:

\bar{E} = E / (C_{w f} T_{0})

(29)

Equations (10)–(12) and (16)–(29) are combined, and the four dimensionless performance indicators of the cycle are obtained as follows:

\bar{W} = \frac{\begin{array}{l} C_{w f} x y (C_{H 1} E_{H 1} T_{H 3} + C_{L \min} E_{L} T_{L 1}) + C_{H \min} E_{H} T_{H 1} {x y [C_{w f} \\ - C_{H 1} E_{H 1} + C_{L \min} E_{L} (η_{t} - 1)] - C_{L \min} E_{L} η_{t}} + a_{1} {C_{L \min} E_{L} \\ \times [(η_{t} - 1) x y - η_{t}] (C_{w f} - E_{H} C_{H \min}) - x y [C_{w f} C_{H \min} E_{H} \\ + C_{H 1} E_{H 1} (C_{w f} - C_{H \min} E_{H})]} \end{array}}{C_{w f}^{2} T_{0} x y}

(30)

η = \frac{\begin{array}{l} C_{H \min} C_{L \min} E_{H} E_{L} η_{t} T_{H 1} - {C_{H \min} E_{H} T_{H 1} [C_{w f} - C_{H 1} E_{H 1} + C_{L \min} E_{L} (η_{t} - 1)] \\ \begin{array}{l} + C_{w f} x y (C_{H 1} E_{H 1} T_{H 3} + C_{L \min} E_{L} T_{L 1})} + a_{1} {[C_{H \min} C_{w f} E_{H} + C_{H 1} E_{H 1} (C_{w f} \\ - E_{H} C_{H \min})] x y - C_{L \min} E_{L} (C_{w f} - C_{H \min} E_{H}) [(η_{t} - 1) x y - η_{t}]} \end{array} \end{array}}{\begin{array}{l} x y {a_{1} [C_{H 1} C_{w f} E_{H 1} + C_{H \min} E_{H} (C_{w f} - C_{H 1} E_{H 1})] + C_{H \min} E_{H} (C_{H 1} E_{H 1} \\ - C_{w f}) T_{H 1} - C_{H 1} C_{w f} E_{H 1} T_{H 3}} \end{array}}

(31)

\bar{P} = \frac{\begin{array}{l} {a_{1} (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L}) [x y (η_{t} - 1) - η_{t}] - C_{L \min} C_{w f} E_{L} T_{L 1} x \\ \times y + E_{H} C_{H \min} T_{H 1} (C_{w f} - C_{L \min} E_{L}) [(η_{t} - 1) x y - η_{t}]} {C_{w f} x y (C_{H 1} E_{H 1} T_{H 3} \\ \begin{array}{l} + C_{L \min} E_{L} T_{L 1}) + {x y [C_{w f} - C_{H 1} E_{H 1} + C_{L \min} E_{L} (η_{t} - 1)] - C_{L \min} E_{L} η_{t}} C_{H \min} \\ \times E_{H} T_{H 1} + a_{1} {C_{L \min} (C_{w f} - E_{H} C_{H \min}) E_{L} {(η_{t} - 1) x y - η_{t} - x y [C_{H \min} C_{w f} E_{H} \\ + C_{H 1} E_{H 1} (C_{w f} - C_{H \min} E_{H})]}}} \end{array} \end{array}}{C_{w f}^{3} T_{0} x y [a_{1} (C_{w f} - C_{H \min} E_{H}) + C_{H \min} E_{H} T_{H 1}] [(η_{t} - 1) x y - η_{t}]}

(32)

\bar{E} = \frac{\begin{array}{l} {C_{w f} x y (C_{H 1} E_{H 1} T_{H 3} + C_{L \min} E_{L} T_{L 1}) + C_{H \min} E_{H} T_{H 1} {x y [C_{w f} - C_{H 1} E_{H 1} \\ + C_{L \min} E_{L} (η_{t} - 1)] - C_{L \min} E_{L} η_{t}} + a_{1} {C_{L \min} E_{L} (C_{w f} - C_{H \min} E_{H}) [(η_{t} \\ - 1) x y - η_{t}] - x y [C_{w f} C_{H \min} E_{H} + C_{H 1} E_{H 1} (C_{w f} - C_{H \min} E_{H})]}} / (T_{0} \\ \begin{array}{l} \times x y) - C_{w f} {C_{L} \ln {1 + {C_{L \min} E_{L} {a_{1} C_{w f} η_{t} - C_{w f} x y [a_{1} (η_{t} - 1) + T_{L 1}] \\ + C_{H \min} E_{H} (a_{1} - T_{H 1}) [(η_{t} - 1) x y - η_{t}]}} / (C_{L} C_{w f} T_{L 1} x y)} + C_{H} \ln {[a_{1} \\ \times C_{H \min} E_{H} + (C_{H} - C_{H \min} E_{H}) T_{H 1}] / (C_{H} T_{H 1})} + C_{H 1} \ln {1 + {E_{H 1} [C_{w f} \\ \times (a_{1} - T_{H 3}) + E_{H} C_{H \min} (T_{H 1} - a_{1})]} / (C_{w f} T_{H 3})}} \end{array} \end{array}}{C_{w f}^{2}}

(33)

where

a_{1} = \frac{(η_{c} + x - 1) {C_{L m i n} C_{w f} E_{L} T_{L 1} x y - C_{H m i n} E_{H} T_{H 1} (C_{w f} - C_{L m i n} E_{L}) [(η_{t} - 1) x y - η_{t}]}}{\begin{array}{l} C_{H m i n} C_{L m i n} E_{H} E_{L} (η_{c} + x - 1) (η_{t} x y - x y - η_{t}) + C_{w f}^{2} [x y - x^{2} y + η_{t} (η_{c} + x \\ - 1) (x y - 1)] - C_{w f} (η_{c} + x - 1) (E_{H} C_{H m i n} + E_{L} C_{L m i n}) [(η_{t} - 1) x y - η_{t}] \end{array}}

(34)

Parameters

x

and

y

in Equations (30)–(34) can be obtained by Equations (13) and (19), and then the arithmetic solution of

\bar{W}

η

\bar{P}

and

\bar{E}

can be gained. When

C_{H}

C_{H 1}

C_{L}

E_{H}

E_{H 1}

E_{L}

η_{c}

and

η_{t}

are specific values, the cycle could be transformed into different cycle models. Equations (30)–(34) could be simplified into the performance indicators of the various cycle models, which have certain universality.

When $C_{H 1} = C_{L} \to \infty$ , Equations (30)–(34) can be simplified into the performance indicators of the irreversible simple BCY with an IHP and coupled to constant-temperature heat reservoirs (CTHRs) whose $T - s$ diagram is shown in Figure 3a:

$\bar{W} = \frac{\begin{array}{l} C_{w f} x y (C_{H 1} E_{H 1} T_{H 3} + C_{L \min} E_{L} T_{L 1}) + C_{H \min} E_{H} T_{H 1} {x y [C_{w f} \\ - C_{H 1} E_{H 1} + C_{L \min} E_{L} (η_{t} - 1)] - C_{L \min} E_{L} η_{t}} + a_{1} {C_{L \min} E_{L} \\ \times [(η_{t} - 1) x y - η_{t}] (C_{w f} - E_{H} C_{H \min}) - x y [C_{w f} C_{H \min} E_{H} \\ + C_{H 1} E_{H 1} (C_{w f} - C_{H \min} E_{H})]} \end{array}}{C_{w f}^{2} T_{0} x y}$

(35)

$η = \frac{\begin{array}{l} C_{w f} E_{H} E_{L} η_{t} T_{H 1} - {E_{H} T_{H 1} [C_{w f} - C_{H 1} E_{H 1} + C_{w f} E_{L} (η_{t} - 1)] + (C_{H 1} E_{H 1} T_{H 3} \\ + C_{w f} E_{L} T_{L 1})} x y + a_{2} {[C_{w f} E_{H} + C_{H 1} E_{H 1} (1 - E_{H})] x y - C_{w f} E_{L} (1 - E_{H}) \\ \times [- η_{t} + (- 1 + η_{t}) x y]} \end{array}}{x y {a_{2} [C_{H 1} E_{H 1} + E_{H} (C_{w f} - C_{H 1} E_{H 1})] + E_{H} T_{H 1} (C_{H 1} E_{H 1} - C_{w f}) - C_{H 1} E_{H 1} T_{H 3}}}$

(36)

$\bar{P} = \frac{\begin{array}{l} C_{w f} {- E_{L} T_{L 1} x y + a_{2} (1 - E_{H}) (1 - E_{L}) [(η_{t} - 1) x y - η_{t}] + E_{H} T_{H 1} (1 - E_{L}) [(η_{t} \\ - 1) x y - η_{t}]} {x y (C_{H 1} E_{H 1} T_{H 3} + C_{w f} E_{L} T_{L 1}) + E_{H} T_{H 1} {x y [C_{w f} - C_{H 1} E_{H 1} + C_{w f} \\ \times E_{L} (η_{t} - 1)] - C_{w f} E_{L} η_{t}} + a_{2} C_{w f} (1 - E_{H}) E_{L} {(η_{t} - 1) x y - η_{t} - C_{w f} x y [C_{w f} E_{H} \\ + C_{H 1} E_{H 1} (1 - E_{H})]}} \end{array}}{C_{w f}^{3} T_{0} x y [a_{2} (1 - E_{H}) + E_{H} T_{H 1}] [(η_{t} - 1) x y - η_{t}]}$

(37)

$\bar{E} = \frac{\begin{array}{l} {x y (C_{H 1} E_{H 1} T_{H 3} + C_{w f} E_{L} T_{L 1}) + E_{H} T_{H 1} {[C_{w f} - C_{H 1} E_{H 1} + C_{w f} E_{L} (η_{t} - 1)] x y - E_{L} η_{t}} \\ + a_{2} C_{w f} {C_{w f} E_{L} (1 - E_{H}) [(η_{t} - 1) x y - η_{t}] - x y [C_{w f} E_{H} + C_{H 1} (1 - E_{H}) E_{H 1}]}} / (T_{0} x y) \\ \begin{array}{l} - {C_{H} \ln [(a_{2} C_{w f} E_{H} + C_{H} T_{H 1} - C_{w f} E_{H} T_{H 1}) / (C_{H} T_{H 1})] + C_{H 1} \ln {1 + {E_{H 1} [a_{2} + C_{w f} E_{H} \\ \times (T_{H 1} - a_{2}) / C_{w f} - T_{H 3}]} / T_{H 3}} + C_{L} \ln {1 + {C_{w f} E_{L} {a_{2} η_{t} - x y [a_{2} (η_{t} - 1) + T_{L 1}] + E_{H} \\ \times (a_{2} - T_{H 1}) [(η_{t} - 1) x y - η_{t}]}} / (C_{L} T_{L 1} x y)}} \end{array} \end{array}}{C_{w f}}$

(38)

where

$a_{2} = \frac{(η_{c} + x - 1) {- E_{L} T_{L 1} x y - E_{H} T_{H 1} (1 - E_{L}) [(η_{t} - 1) x y - η_{t}]}}{\begin{array}{l} E_{H} E_{L} (η_{c} + x - 1) [(x y - 1) η_{t} - x y] + [x y - x^{2} y + η_{t} (η_{c} + x - 1) \\ \times (x y - 1)] - [(η_{t} - 1) x y - η_{t}] (η_{c} + x - 1) (E_{H} + E_{L}) \end{array}}$

(39)
When $η_{c 1} = η_{t 1} = 1$ , Equations (30)–(34) can be respectively simplified into the performance indicators of the endoreversible simple BCY with an IHP and coupled to VTHRs [99], whose $T - s$ diagram is shown in Figure 3b:

$\bar{W} = \frac{\begin{array}{l} C_{w f} x {C_{L \min} C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [C_{w f} T_{H 3} (y - 1) + C_{L \min} E_{L} (T_{H 3} \\ - T_{L 1} x y)]} + E_{H} C_{H \min} {C_{L \min} E_{L} [C_{w f} T_{H 1} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} \\ \times E_{H 1} x (T_{L 1} x y - T_{H 3})] + x C_{w f} [(y - 1) C_{w f} T_{H 1} + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)]} \end{array}}{C_{w f} T_{0} x [C_{w f}^{2} y - (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L})]}$

(40)

$η = \frac{\begin{array}{l} C_{w f} T_{0} x {C_{L \min} C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [C_{w f} T_{H 3} (y - 1) + C_{L \min} E_{L} (T_{H 3} - T_{L 1} x y)]} \\ \begin{array}{l} + C_{H \min} E_{H} {C_{L \min} E_{L} [C_{w f} T_{H 1} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} E_{H 1} x (T_{L 1} x y - T_{H 3})] \\ + C_{w f} x [C_{w f} T_{H 1} (y - 1) + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)]} \end{array} \end{array}}{\begin{array}{l} C_{w f} T_{0} x {C_{H \min} E_{H} [C_{w f}^{2} T_{H 1} (y - 1) + C_{H 1} C_{w f} E_{H 1} (T_{H 3} - T_{H 1} y) + C_{L \min} C_{w f} E_{L} (T_{H 1} - T_{L 1} x y) \\ + C_{H 1} C_{L \min} E_{H 1} E_{L} (T_{L 1} x y - T_{H 3})] + C_{H 1} C_{w f} E_{H 1} [C_{w f} T_{H 3} (y - 1) + C_{L \min} E_{L} (T_{H 3} - T_{L 1} x y)]} \end{array}}$

(41)

$\bar{P} = \frac{\begin{array}{l} [C_{H \min} E_{H} T_{H 1} (C_{w f} - C_{L \min} E_{L}) + C_{L \min} C_{w f} E_{L} T_{L 1} x y] {C_{w f} x {C_{L \min} C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} \\ \begin{array}{l} \times [C_{w f} T_{H 3} (y - 1) + C_{L \min} E_{L} (T_{H 3} - T_{L 1} x y)]} + C_{H \min} E_{H} {C_{w f} x [C_{w f} T_{H 1} (y - 1) + (T_{H 3} - T_{H 1} \\ \times y) C_{H 1} E_{H 1}] + C_{L \min} E_{L} [C_{w f} T_{H 1} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} E_{H 1} x (T_{L 1} x y - T_{H 3})]}} \end{array} \end{array}}{\begin{array}{l} C_{w f} T_{0} x [C_{w f}^{2} y - (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L})] [C_{L \min} (C_{w f} - C_{H \min} E_{H}) E_{L} \\ \times T_{L 1} x + C_{H \min} C_{w f} E_{H} T_{H 1}] \end{array}}$

(42)

$\bar{E} = \frac{\begin{array}{l} C_{w f} x {C_{L \min} C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [C_{w f} T_{H 3} (y - 1) + C_{L \min} E_{L} (T_{H 3} \\ - T_{L 1} x y)]} + C_{H \min} E_{H} {C_{L \min} E_{L} [C_{w f} T_{H 1} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} \\ \times E_{H 1} x (T_{L 1} x y - T_{H 3})] + C_{w f} x [C_{w f} T_{H 1} (y - 1) + (T_{H 3} - T_{H 1} y) C_{H 1} E_{H 1}]} \end{array}}{\begin{matrix} C_{w f} T_{0} x [C_{w f}^{2} y - (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L})] \\ - \frac{C_{H}}{C_{w f} T_{0}} \ln {1 + \frac{\begin{array}{l} C_{H \min} C_{w f} E_{H} (C_{w f} T_{H 1} - C_{L \min} E_{L} T_{H 1} - C_{w f} T_{H 1} y + C_{L \min} \\ \times E_{L} T_{L 1} x y) \end{array}}{T_{H 1} [C_{H} C_{w f}^{2} y - C_{H} (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L})]}} \\ - \frac{C_{H 1}}{C_{w f} T_{0}} \ln \frac{\begin{array}{l} {(C_{w f} - C_{H \min} E_{H}) (E_{H 1} - 1) (C_{w f} - C_{L \min} E_{L}) T_{H 3} + C_{w f} y \\ \times [C_{H \min} E_{H} E_{H 1} T_{H 1} - C_{w f} (E_{H 1} - 1) T_{H 3}] + C_{L \min} E_{H 1} E_{L} T_{L 1} x y \\ \times (C_{w f} - C_{H \min} E_{H})} \end{array}}{C_{w f}^{2} T_{H 3} y - T_{H 3} (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L})} \\ - \frac{C_{L}}{C_{w f} T_{0}} \ln {1 + \frac{C_{L \min} C_{w f} E_{L} [C_{H \min} E_{H} (T_{H 1} - T_{L 1} x) - C_{w f} T_{L 1} x (y - 1)]}{C_{L} T_{L 1} [(C_{H \min} E_{H} - C_{w f}) (C_{w f} C_{L \min} E_{L}) x + C_{w f}^{2} x y]}} \end{matrix}}$

(43)
When $η_{c 1} = η_{t 1} = 1$ and $C_{H 1} = C_{H 2} = C_{L} \to \infty$ , Equations (30)–(34) can be simplified into the performance indicators of the endoreversible simple BCY with an IHP and coupled to CTHRs, whose $T - s$ diagram is shown in Figure 3c:

$\bar{W} = \frac{\begin{array}{l} C_{w f} x {C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [E_{L} T_{H 3} - E_{L} T_{L 1} x y + T_{H 3} (y - 1)]} \\ + C_{w f} E_{H} {E_{L} [T_{H 1} C_{w f} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} E_{H 1} x (T_{L 1} x y \\ - T_{H 3})] + x [C_{w f} T_{H 1} (y - 1) + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)]} \end{array}}{C_{w f}^{2} T_{0} x (E_{H} + E_{L} + y - E_{H} E_{L} - 1)}$

(44)

$η = \frac{\begin{array}{l} T_{0} x {C_{w f} E_{L} T_{L 1} y - C_{w f} E_{L} T_{L 1} + C_{H 1} E_{H 1} [T_{H 3} y - T_{H 3} + T_{H 3} E_{L} - E_{L} T_{L 1} x y]} \\ \begin{array}{l} + {E_{H} {x [C_{w f} T_{H 1} y - C_{w f} T_{H 1} + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)] + E_{L} [C_{w f} T_{H 1} x - C_{w f} \\ \times T_{H 1} + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} E_{H 1} x (- T_{H 3} + T_{L 1} x y)]} \end{array} \end{array}}{\begin{array}{l} C_{w f} T_{0} x {[C_{w f} T_{H 1} y - C_{w f} T_{H 1} + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y) + C_{w f} E_{L} (T_{H 1} - T_{L 1} x y) \\ + C_{H 1} E_{H 1} E_{L} (T_{L 1} x y - T_{H 3})] E_{H} + C_{H 1} E_{H 1} [T_{H 3} (y - 1) + E_{L} (T_{H 3} - T_{L 1} x y)]} \end{array}}$

(45)

$\bar{P} = \frac{\begin{array}{l} [E_{H} T_{H 1} (1 - E_{L}) + E_{L} T_{L 1} x y] {C_{w f} x {C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [T_{H 3} (y - 1) \\ + E_{L} (T_{H 3} - T_{L 1} x y)]} + E_{H} C_{w f} {x [C_{w f} T_{H 1} (y - 1) + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)] \\ + E_{L} [C_{w f} T_{H 1} (x - 1) + C_{w f} T_{L 1} x (1 - x y) + C_{H 1} x E_{H 1} (T_{L 1} x y - T_{H 3})]}} \end{array}}{C_{w f}^{2} T_{0} x (E_{H} + E_{L} + y - E_{H} E_{L} - 1) (E_{H} T_{H 1} + E_{L} T_{L 1} x - E_{L} T_{L 1} x E_{H})}$

(46)

$\bar{E} = \frac{\begin{array}{l} {C_{w f} E_{L} T_{L 1} (y - 1) + C_{H 1} E_{H 1} [T_{H 3} (y - 1) + E_{L} T_{H 3} T_{L 1} x y]} x \\ + E_{H} {E_{L} [C_{w f} T_{H 1} (x - 1) + (1 - x y) C_{w f} T_{L 1} x + C_{H 1} E_{H 1} x (T_{L 1} \\ \times x y - T_{H 3})] + x [C_{w f} T_{H 1} (y - 1) + C_{H 1} E_{H 1} (T_{H 3} - T_{H 1} y)]} \end{array}}{\begin{array}{c} C_{w f} T_{0} x y - C_{w f} T_{0} x (1 - E_{H} - E_{L} + E_{H} E_{L}) \\ - \frac{C_{H 1}}{C_{w f} T_{0}} \ln \frac{\begin{array}{l} C_{w f} T_{H 3} (1 - E_{H} - E_{L} + E_{H} E_{L}) (E_{H 1} - 1) + C_{w f} \\ \times y [E_{H} E_{H 1} T_{H 1} - T_{H 3} (E_{H 1} - 1)] + E_{H 1} E_{L} T_{L 1} x y \\ \times (1 - E_{H}) \end{array}}{C_{w f} [T_{H 3} y - T_{H 3} (1 - E_{H}) (1 - E_{L})]} \\ - \frac{C_{H}}{C_{w f} T_{0}} \ln {1 + \frac{E_{H} C_{w f} (T_{H 1} - E_{L} \times T_{H 1} - T_{H 1} y + E_{L} T_{L 1} x y)}{T_{H 1} C_{H} [y - (1 - E_{H}) (1 - E_{L})]}} \\ - \frac{C_{L}}{C_{w f} T_{0}} \ln {1 + \frac{E_{L} C_{w f} [E_{H} (T_{H 1} - T_{L 1} x) - T_{L 1} x (y - 1)]}{C_{L} T_{L 1} [(E_{H} - 1) (1 - E_{L}) x + x y]}} \end{array}}$

(47)
When $E_{H 1} = 0$ , Equations (30)–(34) can be simplified into the performance indicators of the simple irreversible BCY coupled to VTHRs [79], whose $T - s$ diagram is shown in Figure 3d:

$\bar{W} = \frac{\begin{array}{l} C_{L \min} C_{w f} E_{L} T_{L 1} x + C_{H \min} E_{H} T_{H 1} {C_{L \min} E_{L} [η_{t} (x - 1) - x] + C_{w f} x} \\ + a_{3} {C_{L \min} (C_{w f} - C_{H \min} E_{H}) E_{L} [η_{t} (x - 1) - x] - C_{H \min} C_{w f} E_{H} x} \end{array}}{C_{w f}^{2} T_{0} x}$

(48)

$η = \frac{\begin{array}{l} a_{3} {C_{H \min} C_{w f} E_{H} x - C_{L \min} E_{L} (C_{w f} - C_{H \min} E_{H}) [η_{t} (x - 1) - x]} - C_{L \min} \\ \times C_{w f} E_{L} T_{L 1} x + C_{H \min} E_{H} T_{H 1} [C_{L \min} E_{L} (η_{t} + x - η_{t} x) - C_{w f} x] \end{array}}{x C_{H \min} E_{H} C_{w f} (a_{3} - T_{H 1})}$

(49)

$\bar{P} = \frac{\begin{array}{l} {- a_{3} [η_{t} (x - 1) - x] (C_{w f} - C_{H \min} E_{H}) (C_{w f} - C_{L \min} E_{L}) - C_{H \min} E_{H} T_{H 1} [η_{t} (x - 1) - x] \\ \times (C_{w f} - C_{L \min} E_{L}) + C_{L \min} C_{w f} E_{L} T_{L 1} x} {a_{3} {C_{L \min} E_{L} [η_{t} (x - 1) - x] (C_{w f} - C_{H \min} E_{H}) \\ - C_{H \min} C_{w f} E_{H} x} + C_{H \min} E_{H} T_{H 1} {C_{L \min} E_{L} [η_{t} (x - 1) - x] + C_{w f} x} + C_{L \min} C_{w f} E_{L} T_{L 1} x} \end{array}}{- C_{w f}^{3} T_{0} x [η_{t} (x - 1) - x] [a_{3} (C_{w f} - C_{H \min} E_{H}) + C_{H \min} E_{H} T_{H 1}]}$

(50)

$\bar{E} = \frac{\begin{array}{l} {C_{L \min} C_{w f} E_{L} T_{L 1} x + C_{H \min} E_{H} T_{H 1} {C_{L \min} E_{L} [η_{t} (x - 1) - x] + C_{w f} x} + a_{3} {C_{L \min} (C_{w f} \\ - C_{H \min} E_{H}) E_{L} [η_{t} (x - 1) - x] - C_{H \min} C_{w f} E_{H} x}} / (T_{0} x) - C_{w f} {C_{H} \ln [1 + C_{H \min} E_{H} \\ \times (a_{3} - T_{H 1}) / (C_{H} T_{H 1})] + C_{L} \ln {1 + C_{L \min} E_{L} {a_{3} C_{w f} η_{t} + C_{H \min} E_{H} (a_{3} - T_{H 1}) [η_{t} (x - 1) \\ - x] - C_{w f} [a_{3} (η_{t} - 1) + T_{L 1}] x} / (C_{L} C_{w f} T_{L 1} x)}} \end{array}}{C_{w f}^{2}}$

(51)

where

$a_{3} = \frac{(η_{c} + x - 1) {C_{L m i n} C_{w f} E_{L} T_{L 1} x - C_{H m i n} E_{H} T_{H 1} (C_{w f} - C_{L m i n} E_{L}) [(η_{t} - 1) x - η_{t}]}}{\begin{array}{l} C_{H m i n} C_{L m i n} E_{H} E_{L} (η_{c} + x - 1) (η_{t} x - x - η_{t}) + C_{w f}^{2} [x - x^{2} + η_{t} (η_{c} + x - 1) (x \\ - 1)] - C_{w f} (η_{c} + x - 1) (E_{H} C_{H m i n} + E_{L} C_{L m i n}) \times [(η_{t} - 1) x - η_{t}] \end{array}}$

(52)
When $E_{H 1} = 0$ and $C_{H} = C_{L} \to \infty$ , Equations (30)–(34) can be simplified into the performance indicators of the simple irreversible BCY coupled to CTHRs [76], whose $T - s$ diagram is shown in Figure 3e:

$\bar{W} = \frac{E_{L} T_{L 1} x - a_{4} {(E_{H} - 1) E_{L} [η_{t} (x - 1) - x] + E_{H} x} + E_{H} T_{H 1} [E_{L} η_{t} (x - 1) + x - E_{L} x]}{T_{0} x}$

(53)

$η = \frac{\begin{array}{l} a_{4} (E_{H} - 1) E_{L} [η_{t} (x - 1) - x] + a_{4} E_{H} x - E_{H} T_{H 1} x - E_{L} T_{L 1} x \\ + E_{H} E_{L} T_{H 1} (η_{t} + x - η_{t} x) \end{array}}{x E_{H} (a_{4} - T_{H 1})}$

(54)

$\bar{P} = \frac{\begin{array}{l} {a_{4} (E_{H} - 1) (E_{L} - 1) [η_{t} (x - 1) - x] - E_{H} T_{H 1} (E_{L} - 1) [η_{t} (x \\ - 1) - x] - E_{L} T_{L 1} x} {a_{4} (E_{H} - 1) E_{L} [η_{t} (x - 1) - x] + a_{4} E_{H} x \\ - E_{H} T_{H 1} x - E_{L} T_{L 1} x + E_{H} E_{L} T_{H 1} (η_{t} + x - η_{t} x)} \end{array}}{T_{0} [a_{4} (E_{H} - 1) - E_{H} T_{H 1}] [η_{t} (x - 1) - x] x}$

(55)

$\begin{array}{l} \bar{E} = {E_{L} T_{L 1} x - a_{4} {E_{L} (E_{H} - 1) [η_{t} (x - 1) - x] + E_{H} x} + E_{H} T_{H 1} [E_{L} η_{t} (x \\ - 1) + x - E_{L} x]} / (T_{0} x) - C_{H} \ln [1 + C_{w f} E_{H} (a_{4} - T_{H 1}) / (C_{H} T_{H 1})] / C_{w f} \\ - C_{L} \ln {1 + C_{w f} E_{L} {a_{4} (E_{H} - 1) [η_{t} (x - 1) - x] - T_{L 1} x + E_{H} T_{H 1} (η_{t} + x \\ - η_{t} x)} / (C_{L} T_{L 1} x)} / C_{w f} \end{array}$

(56)

where

$a_{4} = \frac{(η_{c} + x - 1) E_{H} T_{H 1} (E_{L} - 1) [η_{t} (x - 1) - x] + E_{L} T_{L 1} x}}{(E_{H} - 1) (E_{L} - 1) (x - 1) (η_{c} + x - 1) η_{t} - x [x - 1 + E_{H} (E_{L} - 1) (η_{c} + x - 1) - E_{L} (η_{c} + x - 1)]}$

(57)
When $E_{H 1} = 0$ and $η_{c} = η_{t} = 1$ , Equations (30)–(34) can be simplified into the performance indicators of the simple endoreversible BCY coupled to VTHRs [78], whose $T - s$ diagram is shown in Figure 3f:

$\bar{W} = \frac{C_{H \min} C_{L \min} E_{H} E_{L} (- 1 + x) (T_{H 1} - T_{L 1} x)}{T_{0} x [C_{L \min} C_{w f} E_{L} + C_{H \min} E_{H} (C_{w f} - C_{L \min} E_{L})]}$

(58)

$η = (x - 1) / x$

(59)

$\bar{P} = \frac{\begin{array}{l} C_{H \min} C_{L \min} E_{H} E_{L} (- 1 + x) (T_{H 1} - T_{L 1} x) [C_{H \min} E_{H} (C_{w f} \\ - C_{L \min} E_{L}) T_{H 1} + C_{L \min} C_{w f} E_{L} T_{L 1} x] \end{array}}{\begin{array}{l} T_{0} x [C_{L \min} C_{w f} E_{L} + C_{H \min} E_{H} (C_{w f} - C_{L \min} E_{L})] [C_{L \min} C_{w f} \\ \times E_{L} T_{L 1} x + C_{H \min} E_{H} (C_{w f} T_{H 1} - C_{L \min} E_{L} T_{L 1} x)] \end{array}}$

(60)

$\bar{E} = \frac{\begin{array}{l} \frac{C_{H m i n} C_{L m i n} C_{w f} E_{H} E_{L} (x - 1) (T_{H 1} - T_{L 1} x)}{\begin{array}{l} [C_{L m i n} C_{w f} E_{L} + C_{H m i n} E_{H} (C_{w f} \\ - C_{L m i n} E L)] T_{0} x \end{array}} - C_{H} \ln [1 + \frac{C_{H m i n} C_{L m i n} C_{w f} E_{H} E_{L} (T_{L 1} x - T_{H 1})}{\begin{array}{l} C_{H} [C_{L m i n} C_{w f} E_{L} + C_{H m i n} E_{H} \\ (C_{w f} - C_{L m i n} E_{L})] T_{H 1} \end{array}}] \\ - C_{L} \ln {\frac{C_{L} C_{L m i n} C_{w f} E_{L} T_{L 1} x + C_{H m i n} E_{H} [C_{L} C_{w f} T_{L 1} x + C_{L m i n} E_{L} (C_{w f} T_{H 1} - C_{L} T_{L 1} x - C_{w f} T_{L 1} x)]}{C_{L} [C_{L m i n} C_{w f} E_{L} + C_{H m i n} E_{H} (C_{w f} - C_{L m i n} E_{L})] T_{L 1} x}} \end{array}}{C_{w f}}$

(61)
When $E_{H 1} = 0$ , $η_{c} = η_{t} = 1$ and $C_{H} = C_{L} \to \infty$ , Equations (30)–(34) can be simplified into the performance indicators of the simple endoreversible BCY coupled to CTHRs [77], whose $T - s$ diagram is shown in Figure 3g:

$\bar{W} = \frac{E_{H} E_{L} (- 1 + x) (T_{L 1} x - T_{H 1})}{[E_{H} (E_{L} - 1) - E_{L}] T_{0} x}$

(62)

$η = (x - 1) / x$

(63)

$\bar{P} = \frac{E_{H} E_{L} (x - 1) (T_{L 1} x - T_{H 1}) [E_{H} (E_{L} - 1) T_{H 1} - E_{L} T_{L 1} x]}{T_{0} x (E_{H} E_{L} T_{L 1} x - E_{H} T_{H 1} - E_{L} T_{L 1} x) [E_{H} (E_{L} - 1) - E_{L}]}$

(64)

$\bar{E} = \frac{\begin{array}{l} C_{w f} E_{H} E_{L} (x - 1) (T_{L 1} x - T_{H 1}) + C_{H} T_{0} x (E_{H} + E_{L} - E_{H} E_{L}) \ln {1 - C_{w f} E_{H} \\ \times E_{L} (T_{H 1} - T_{L 1} x) / [C_{H} T_{H 1} (E_{H} + E_{L} - E_{H} E_{L})]} + C_{L} T_{0} x (E_{H} + E_{L} - E_{H} \\ \times E_{L}) \ln [1 + C_{w f} E_{H} E_{L} (T_{H 1} - T_{L 1} x) / (C_{L} (E_{H} + E_{L} - E_{H} E_{L}) T_{L 1} x)] \end{array}}{C_{w f} [E_{H} (E_{L} - 1) - E_{L}] T_{0} x}$

(65)
When $E_{H} = E_{L} = 0$ , $η_{c} = η_{t} = 1$ and $C_{w f} \to \infty$ , the cycle in this paper can become the endoreversible Carnot cycle coupled to VTHRs [14], whose $T - s$ diagram is shown in Figure 3h. However, Equations (30), (33), and (34) need to be de-dimensionalized to simplify to $W$ , $P$ and $E$ of the endoreversible Carnot cycle coupled to VTHRs. The performance indicators of the cycle are:

$W = \frac{C_{H} C_{L} E_{H} E_{L} (x - 1) (T_{H 1} - T_{L 1} x)}{x (C_{H} E_{H} + C_{L} E_{L})}$

(66)

$η = (x - 1) / x$

(67)

$P = \frac{C_{H} C_{L} E_{H} E_{L} (x - 1) (T_{H 1} - T_{L 1} x)}{x (C_{H} E_{H} + C_{L} E_{L})}$

(68)

$\begin{array}{l} E = \frac{C_{H} C_{L} E_{H} E_{L} (x - 1) (T_{H 1} - T_{L 1} x)}{(C_{H} E_{H} + C_{L} E_{L}) x} - C_{H} T_{0} \ln [1 + \frac{C_{L} E_{H} E_{L} (T_{L 1} x - T_{H 1})}{(C_{H} E_{H} + C_{L} E_{L}) T_{H 1}}] \\ - C_{L} T_{0} \ln [\frac{C_{H} E_{H} E_{L} T_{H 1} + C_{H} E_{H} T_{L 1} x + C_{L} E_{L} T_{L 1} x - C_{H} E_{H} E_{L} T_{L 1} x}{C_{H} E_{H} T_{L 1} x + C_{L} E_{L} T_{L 1} x}] \end{array}$

(69)
When $E_{H} = E_{L} = 0$ , $η_{c} = η_{t} = 1$ and $C_{H 1} = C_{L} = C_{w f} \to \infty$ , the cycle in this paper can become the endoreversible Carnot cycle coupled to CTHRs [12], whose $T - s$ diagram is shown in Figure 3i. However, Equations (30), (33), and (34) also need to be de-dimensionalized to simplify to $W$ , $P$ and $E$ of the cycle [12,74,82]. The performance indicators of the cycle are:

$W = \frac{U_{H} U_{L} (- 1 + x) (T_{H 1} - T_{L 1} x)}{(U_{H} + U_{L}) x}$

(70)

$η = (x - 1) / x$

(71)

$P = \frac{U_{H} U_{L} (- 1 + x) (T_{H 1} - T_{L 1} x)}{(U_{H} + U_{L}) x}$

(72)

$E = \frac{U_{H} U_{L} (T_{H 1} - T_{L 1} x) [(T_{0} + T_{H 1}) T_{L 1} x - T_{H 1} (T_{0} + T_{L 1})]}{T_{H 1} T_{L 1} (U_{H} + U_{L}) x}$

(73)
When $E_{H} = E_{L} = 0$ , $η_{c} = η_{t} = 1$ , $C_{H 1} = C_{L} = C_{w f} \to \infty$ , and $U_{L} \to \infty$ , the cycle in this paper can become the endoreversible Novikov cycle coupled to CTHRs [11], whose $T - s$ diagram is shown in Figure 3j. However, Equations (30), (33), and (34) also need to be de-dimensionalized to simplify to $W$ , $P$ and $E$ of the cycle [11]. The performance indicators of the cycle are:

$W = \frac{U_{H} (x - 1) (T_{H 1} - T_{L 1} x)}{x}$

(74)

$η = (x - 1) / x$

(75)

$P = \frac{U_{H} (x - 1) (T_{H 1} - T_{L 1} x)}{x}$

(76)

$E = \frac{U_{H} (T_{H 1} - T_{L 1} x) [T_{H 1} T_{L 1} (x - 1) + T_{0} (T_{L 1} x - T_{H 1})]}{T_{H 1} T_{L 1} x}$

(77)
Through comparison with the results in Refs [11,12,13,14,59,76,77,78,79,99], it is found that the results of this paper are consistent with those in Refs [11,12,13,14,59,76,77,78,79,99], which further illustrates the accuracy of the model established in this paper. In particular, when the powers in Equations (58), (62), (66), (70), and (74) take the maximum values, namely $x = \sqrt{T_{H 1} / T_{L 1}}$ , the efficiencies at the maximum power point, Equations (59), (63), (67), (71), and (75) are $η = 1 - \sqrt{T_{L 1} / T_{H 1}}$ , which was derived in Refs. [10,11,12] by Moutier [10], Novikov [11], and Curzon and Ahlborn [12]. One can see that the results of this paper include the Novikov–Curzon–Ahlborn efficiency.
FTT is the further extension of conventional irreversible thermodynamics. The cycle model established by Curzon and Ahlborn [12] was a reciprocating Carnot cycle, and the finite time was its major feature. The methods used for solving the FTT problem are usually variational principle and optimal control theory. Therefore, such problems of extremal of thermodynamic processes were first named as FTT by Andresen et al. [132] and as Optimization Thermodynamics or Optimal Control in Problems of Extremals of Irreversible Thermodynamic Processes by Orlov and Rudenko [133]. When the research object was extended from reciprocating devices characterized by finite-time to the steady state flow devices characterized by finite-size, one realizes that the physical property of the problems is the heat transfer owing to temperature deference. Therefore, Grazzini [14] termed it Finite Temperature Difference Thermodynamics, and Lu [134] termed it Finite Surface Thermodynamics. In fact, the works performed by Moutier [10] and Novikov [11] were also steady state flow device models. Bejan introduced the effect of temperature difference heat transfer on the total entropy generation of the systems, taking the entropy generation minimization as the optimization objective for designing thermodynamic processes and devices, termed “Entropy Generation Minimization” or “Thermodynamic Optimization” [15,135]. For the steady state flow device models, Feidt [136,137,138,139,140,141,142,143,144,145,146] termed it Finite Physical Dimensions Thermodynamics (FPDT). The model established herein is closer to FPDT. For both reciprocating model and steady state flow model, the suitable name may be thermodynamics of finite size devices and finite time processes, as Bejan termed it [15,135]. According to the idiomatic usage, the theory is termed FTT in this paper.

3. Analyses and Optimizations with Each Single Objective

3.1. Analyses of Each Single Objective

The impacts of the irreversibility on cycle performance indicators (

\bar{W}

η

\bar{P}

and

\bar{E}

) are analyzed below. In numerical calculations, it is set that

C_{L} = C_{H} = 1.2 kW / K

C_{w f} = 1 kW / K

T_{0} = 300 K

C_{H 1} = 0.6 kW / K

k = 1.4

R_{g} = 0.287 kJ / (kg \cdot K)

E_{H} = E_{H 1} = E_{L} = 0.9

C_{p} = 1.005 kW / K

τ_{H 1} = 4.33

τ_{H 3} = 5

and

τ_{L} = 1

Figure 4, Figure 5 and Figure 6 present the relationships of

\bar{W}

η

\bar{P}

\bar{E}

π_{t}

and

v_{5} / v_{1}

versus

π

with different

η_{t}

. As shown in Figure 4 and Figure 5,

\bar{W}

η

\bar{P}

and

\bar{E}

increase and then decrease as

π

increases. In the same situation,

\bar{W}

\bar{E}

\bar{P}

and

η

reach the maximum value successively. When

η_{t} = 0.7

and

π = 32.3

\bar{W} = \bar{P} = 0

. If

π

keeps going up,

\bar{W}

and

\bar{P}

are going to go negative.

\bar{W}

η

\bar{P}

and

\bar{E}

increase as

η_{t}

increases. As

π

increases,

\bar{W}

η

\bar{P}

and

\bar{E}

are affected more significantly by

η_{t}

. As shown in Figure 6,

π_{t}

goes up but

v_{5} / v_{1}

goes down as

π

goes up.

π_{t}

and

v_{5} / v_{1}

decrease as

η_{t}

rises. It illustrates that the degree of the IHP is improved and the device’s volume is reduced as

η_{t}

increases.

By numerical calculations, the influences of

η_{c}

\bar{W}

η

\bar{P}

\bar{E}

and

π_{t}

are the same as those of

η_{t}

\bar{W}

η

\bar{P}

\bar{E}

and

π_{t}

. When

η_{t} = 0.7

and

π = 32.8

\bar{W} = \bar{P} = 0

. However, the impacts of

η_{c}

\bar{W}

η

\bar{P}

and

\bar{E}

are less than those of

η_{t}

\bar{W}

η

\bar{P}

\bar{E}

. The effect of

η_{c}

π_{t}

is more significant than that of

η_{t}

π_{t}

η_{c}

has little effect on

v_{5} / v_{1}

. In the actual design process, it is suggested that

η_{t}

should be given priority.

To further explain the difference between the models in this paper and Ref. [101], the comparison of

\bar{W}

under the variable and constant

π

is shown in Figure 7. As shown in Figure 7,

\bar{W}

increases and then decreases as

π

increases in both cases; that is, the qualitative law is the same. However, there is an apparent quantitative difference between the two points. Under the constant

π

\bar{W}

corresponding to the constant

π

is always greater than

\bar{W}

conforming to the variable

π

. Similarly, there are quantitative differences in

η

\bar{P}

and

\bar{E}

under the variable and constant

π

. The model whose

π

is variable is more realistic.

3.2. Performance Optimizations for Each Single Objective

With four performance indicators as the OPOs, respectively, the HCDs are optimized under the condition of given total heat conductance (

U_{T}

). The optimal results under different OPOs are compared. The HCDs among the RCC, CCC, and precooler are:

u_{H} = U_{H} / U_{T}, u_{H 1} = U_{H 1} / U_{T}, u_{L} = U_{L} / U_{T}

(78)

The HCDs are must larger than 0, the sum of them is 1, and

2 \leq π \leq 50

Figure 8 shows the flowchart of HCD optimization. The steps are as follows:

Enter the known data and the initial values of the HCDs.
The $π_{t}$ is calculated according to Equation (13).
Judge whether the $π_{t}$ $π$ and HCDs meet the constraints. If they are satisfied, perform step 4; if they are not satisfied, go back to step 1.
The performance indicator is solved.
Determine whether the inverse objective function is minimized by using the “fmincon” in MATLAB. If it is the smallest, perform step 6; if it is not the slightest, go back to step 1.
Calculate the other thermodynamic parameters, and the maximum of the performance indicator is obtained.

3.2.1. Optimizations of Each Single Objective

The optimization results of four performance indicators are similar. The optimization results with

η

as the performance indicator will be mainly discussed herein, while the results with

\bar{W}

\bar{P}

and

\bar{E}

as the performance indicators are briefly discussed. The relationships of the optimal thermal efficiency (

η_{opt}

) and the corresponding dimensionless power output (

{\bar{W}}_{η_{opt}}

) versus

π

are shown in Figure 9. The relationships of the corresponding dimensionless power density (

{\bar{P}}_{η_{opt}}

) and the corresponding dimensionless ecological function (

{\bar{E}}_{η_{opt}}

) versus

π

are demonstrated in Figure 10. As shown in Figure 9 and Figure 10,

{\bar{W}}_{η_{opt}}

η_{opt}

{\bar{P}}_{η_{opt}}

and

{\bar{E}}_{η_{opt}}

first rise and then drops as

π

rises, which indicates a parabolic relationship with the downward opening. The corresponding isothermal pressure drop ratio (

{(π_{t})}_{η_{opt}}

) and dimensionless maximum specific volume (

{(v_{5} / v_{1})}_{η_{opt}}

) versus

π

are shown in Figure 11.

{(π_{t})}_{η_{opt}}

decreases and then increases as

π

increases. It indicates that there is a

π_{t}

that maximizes the degree of isothermal heating in the cycle.

{(v_{5} / v_{1})}_{η_{opt}}

decreases as

π

increases. The relationships of the HCDs (

{(u_{H})}_{η_{opt}}

{(u_{H 1})}_{η_{opt}}

and

{(u_{L})}_{η_{opt}}

) versus

π

are shown in Figure 12. As

π

increases,

{(u_{H})}_{η_{opt}}

decreases,

{(u_{H 1})}_{η_{opt}}

increases rapidly and then slowly, and

{(u_{L})}_{η_{opt}}

decreases first and then increases gradually.

By numerical calculations,

{\bar{W}}_{opt}

η_{{\bar{W}}_{opt}}

{\bar{P}}_{{\bar{W}}_{opt}}

{\bar{E}}_{{\bar{W}}_{opt}}

{\bar{W}}_{{\bar{P}}_{opt}}

η_{{\bar{P}}_{opt}}

{\bar{P}}_{opt}

{\bar{E}}_{{\bar{P}}_{opt}}

{\bar{W}}_{{\bar{E}}_{opt}}

η_{{\bar{E}}_{opt}}

{\bar{P}}_{{\bar{E}}_{opt}}

and

{\bar{E}}_{opt}

increase first and then decrease as

π

increases. As

π

increases,

{(π_{t})}_{{\bar{W}}_{opt}}

{(π_{t})}_{{\bar{P}}_{opt}}

and

{(π_{t})}_{{\bar{E}}_{opt}}

reduce first and then increase, and

{(π_{t})}_{{\bar{W}}_{opt}}

{(π_{t})}_{{\bar{E}}_{opt}}

{(π_{t})}_{η_{opt}}

and

{(π_{t})}_{{\bar{P}}_{opt}}

reached the minimum successively. As

π

increases,

{(v_{5} / v_{1})}_{{\bar{W}}_{opt}}

, and

{(v_{5} / v_{1})}_{{\bar{E}}_{opt}}

decline, and their values have little difference.

{(u_{H})}_{{\bar{W}}_{opt}}

{(u_{H})}_{η_{opt}}

{(u_{H})}_{{\bar{P}}_{opt}}

and

{(u_{H})}_{{\bar{E}}_{opt}}

decrease as

π

increases, and

{(u_{H})}_{η_{opt}}

is always the smallest.

{(u_{H 1})}_{{\bar{W}}_{opt}}

and

{(u_{H 1})}_{{\bar{E}}_{opt}}

rise firstly and then tend to keep constant as

π

rises.

{(u_{H 1})}_{{\bar{P}}_{opt}}

first increases then decreases and finally tends to stay stable as

π

rises.

{(u_{L})}_{{\bar{W}}_{opt}}

{(u_{L})}_{{\bar{P}}_{opt}}

and

{(u_{L})}_{{\bar{E}}_{opt}}

first increase rapidly and then slowly as

π

increases.

3.2.2. Influences of Temperature Ratios on Optimization Results

With

η

as the performance indicator, the influences of the temperature ratios on the optimization results are discussed. The relationship of the maximum thermal efficiency (

η_{\max}

) versus

τ_{H 1}

and

τ_{H 3}

is shown in Figure 13. According to Figure 12, the surface is divided into three parts by line

τ_{H 3} = τ_{H 1} + 0.27

(the correlation coefficient is

r_{1} = 0.9969

) and

τ_{H 3} = 1.2 τ_{H 1} + 0.1

(the correlation coefficient is

r_{2} = 1.0000

τ_{H 1}

has little influence on

η_{\max}

. When

τ_{H 3} < 1.2 τ_{H 1} + 0.1

η_{\max}

increases as

τ_{H 3}

increases; when

τ_{H 3} > 1.2 τ_{H 1} + 0.1

τ_{H 3}

has little impact on

η_{\max}

. It is recommended to magnify

τ_{H 1}

By numerical calculations, the surface is divided into three parts by line

τ_{H 3} = 0.84 τ_{H 1} + 0.41

(the correlation coefficient is

r_{1} = 0.9973

) and

τ_{H 3} = 1.2 τ_{H 1} + 0.23

(the correlation coefficient is

r_{2} = 0.9988

) with

\bar{W}

as the performance indicator. The surface is divided into three parts by line

τ_{H 3} = 0.78 τ_{H 1} + 0.6

(the correlation coefficient is

r_{1} = 0.9574

) and

τ_{H 3} = 1.2 τ_{H 1} + 0.33

(the correlation coefficient is

r_{2} = 0.9991

) with

\bar{P}

as the performance indicator. The surface is divided into three parts by line

τ_{H 3} = 0.93 τ_{H 1} + 0.058

(the correlation coefficient is

r_{1} = 0.9978

) and

τ_{H 3} = 1.1 τ_{H 1} + 0.41

(the correlation coefficient is

r_{2} = 0.9990

) with

\bar{E}

as the performance indicator. In practice, the difference between

τ_{H 1}

and

τ_{H 3}

should be controlled and should not be too large.

3.2.3. Influences of the Compressor and the Turbine’s Irreversibilities on Optimization Results

With the four performance indicators as OPOs, respectively, the influences of

η_{c}

and

η_{t}

on optimization results are considered, and the thermodynamic parameters under various optimal performance indicators are compared. Figure 14 and Figure 15 show relationships of

\bar{W}

and

π

under various optimal performance indicators versus

η_{c}

and

η_{t}

, respectively

{\bar{W}}_{\max}

{\bar{P}}_{\max}

, and

{\bar{E}}_{\max}

are the maximum dimensionless power output, maximum dimensionless power density, and maximum dimensionless ecological function, respectively. When

{\bar{W}}_{\max}

η_{\max}

{\bar{P}}_{\max}

, and

{\bar{E}}_{\max}

are used as subscripts, they indicate the corresponding values at

{\bar{W}}_{\max}

η_{\max}

{\bar{P}}_{\max}

, and

{\bar{E}}_{\max}

points.

As shown in Figure 14,

\bar{W}

under various optimal performance indicators increases as

η_{c}

η_{t}

increases. When

η_{c}

and

η_{t}

both approach 1,

{\bar{W}}_{η_{\max}}

first increases and then decreases as

η_{c}

η_{t}

increases. When

η_{c} = η_{t} = 1

η

rises monotonically as

π

gains, and there is no maximum value. In the case of the same

η_{c}

and

η_{t}

, there is

{\bar{W}}_{\max} > {\bar{W}}_{{\bar{E}}_{\max}} > {\bar{W}}_{{\bar{P}}_{\max}} > {\bar{W}}_{η_{\max}}

. As shown in Figure 15,

π

under various optimal performance indicators all increase as

η_{c}

η_{t}

increases. But the influence of

η_{t}

π

is more significant than that of

η_{c}

π

. When

η_{c}

and

η_{t}

both approach 1,

π_{η_{\max}}

is always 50. Because the upper limit of

π

is 50. In the case of the same

η_{c}

and

η_{t}

, there is

π_{η_{\max}} > π_{{\bar{P}}_{\max}} > π_{{\bar{E}}_{\max}} > π_{{\bar{W}}_{\max}}

. The given range of

π

2 \leq π \leq 50

, so when

π = 50

, the trends of

{\bar{W}}_{η_{\max}}

and

π_{η_{\max}}

change significantly.

By numerical calculations,

η

\bar{P}

, and

\bar{E}

under various optimal performance indicators increases as

η_{c}

η_{t}

increases. When

η_{c}

and

η_{t}

both approach 1,

{\bar{P}}_{η_{\max}}

and

{\bar{E}}_{η_{\max}}

first rises and then drops as

η_{c}

η_{t}

rises. In the same

η_{c}

and

η_{t}

, there are

η_{\max} > η_{{\bar{P}}_{\max}} > η_{{\bar{E}}_{\max}} > η_{{\bar{W}}_{\max}}

{\bar{P}}_{\max} > {\bar{P}}_{{\bar{E}}_{\max}} > {\bar{P}}_{{\bar{W}}_{\max}} > {\bar{P}}_{η_{\max}}

, (when

η_{c}

and

η_{t}

both tend to 1, the relationship does not work) and

{\bar{E}}_{\max} > {\bar{E}}_{{\bar{P}}_{\max}} > {\bar{E}}_{{\bar{W}}_{\max}} > {\bar{E}}_{η_{\max}}

(the difference between

{\bar{E}}_{{\bar{P}}_{\max}}

and

{\bar{E}}_{{\bar{W}}_{\max}}

is very small).

The calculations also show that the thermal capacitance rate matchings among the VTHRs and working fluid have influences on the cycle performance.

{\bar{W}}_{\max}

η_{\max}

{\bar{P}}_{\max}

, and

{\bar{E}}_{\max}

increase first and then keep constants as

C_{H} / C_{w f}

C_{H 1} / C_{w f}

increases, and the effects of

C_{H} / C_{w f}

{\bar{W}}_{\max}

η_{\max}

{\bar{P}}_{\max}

, and

{\bar{E}}_{\max}

are more significant than that of

C_{H 1} / C_{w f}

4. Multi-Objective Optimization

4.1. Optimization Algorithm and Decision-Making Methods

It is impossible to achieve the maximums of

\bar{W}

η

\bar{P}

, and

\bar{E}

under the same

π

. It shows that there is a contradiction among the four performance indicators. The multi-objective optimization problem is solved by applying the NSGA-II algorithm [99,100,102,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125]. The detailed optimization process is shown in Figure 16. The Pareto frontier of the cycle performance is obtained by taking

\bar{W}

η

\bar{P}

, and

\bar{E}

as OPOs, using the NSGA-II algorithm. The optimal scheme is selected by using the LINMAP, TOPSIS, and Shannon Entropy methods [99,102], and the algorithm of “gamultiobj” in MATLAB is based on the NSGA-II algorithm. The calculations are assisted by applying the “gamultiobj”, and the corresponding Pareto frontier could be obtained. The parameter settings of “gamultiobj” are listed in Table 1.

The positive and negative ideal points are the optimal and inferior schemes of each performance indicator. The LINMAP method is the Euclidian distance between each scheme and the positive ideal point, among which the one with the smallest distance is the best scheme. Suppose that the Pareto front contains n feasible solutions, and each viable solution contains m objective values

F_{i j}

(

1 \leq i \leq m

and

1 \leq j \leq n

). After normalizing

F_{i j}

, the value

B_{i j}

is:

B_{i j} = F_{i j} / \sqrt{\sum_{i = 1}^{n} F_{i j}^{2}}

(79)

The weight of the j-th OPO is

w_{j}^{LINMAP}

, and the weighted value of

B_{i j}

G_{i j}

G_{i j} = w_{j}^{LINMAP} \cdot B_{i j}

(80)

The j-th objective of the positive ideal point is normalized and weighted, and the corresponding value is

G_{j}^{positive}

. The Euclidean distance between the i-th feasible solution and the positive ideal point is

E D_{i}^{+}

E D_{i}^{+} = \sqrt{\sum_{j = 1}^{m} {(G_{i j} - G_{j}^{positive})}^{2}}

(81)

The best viable solution to the LINMAP method is

i_{opt}

i_{opt} \in \min {E D_{i}^{+}}

(82)

The TOPSIS method considers the Euclidean distance among each scheme and the positive and negative ideal points comprehensively, to further obtain the best scheme. The weight of the j-th OPO is

w_{j}^{TOPSIS}

, and the weighted value of

B_{i j}

G_{i j}

G_{i j} = w_{j}^{TOPSIS} \cdot B_{i j}

(83)

The j-th objective of the negative ideal point is normalized and weighted, and the corresponding value is

G_{j}^{negative}

. The Euclidean distance between the i-th feasible solution and the negative ideal point is

E D_{i}^{-}

E D_{i}^{-} = \sqrt{\sum_{j = 1}^{m} {(G_{i j} - G_{j}^{negative})}^{2}}

(84)

The best feasible solution of the TOPSIS method is

i_{opt}

i_{opt} \in \min {\frac{E D_{i}^{-}}{E D_{i}^{+} + E D_{i}^{-}}}

(85)

The Shannon Entropy method is a method to get the weight of multi-attribute decision-making.

After normalization of

F_{i j}

P_{i j}

is obtained:

P_{i j} = F_{i j} / \sum_{i = 1}^{n} F_{i j}

(86)

The Shannon Entropy and weight of the j-th OPO are:

S E_{j} = - \frac{1}{\ln n} \sum_{i = 1}^{n} P_{i j} \ln P_{i j}

(87)

w_{j}^{Shannon Entropy} = (1 - S E_{j}) / \sum_{j = 1}^{n} (1 - S E_{j})

(88)

The best feasible solution of the TOPSIS method is

i_{opt}

i_{opt} \in \min {P_{i j} \cdot w_{j}^{Shannon Entropy}}

(89)

The deviation index D is defined as:

D = \frac{\sqrt{\sum_{j = 1}^{m} {(G_{i_{opt} j} - G_{j}^{positive})}^{2}}}{\sqrt{\sum_{j = 1}^{m} {(G_{i_{opt} j} - G_{j}^{positive})}^{2}} + \sqrt{\sum_{j = 1}^{m} {(G_{i_{opt} j} - G_{j}^{positive})}^{2}}}

(90)

In this paper,

w_{j}^{LINMAP} = w_{j}^{TOPSIS} = 1

is chosen for the convenience of calculation.

4.2. Multi-Objective Optimization Results

Figure 17 shows the Pareto frontier and optimal schemes corresponding to the four objectives (

\bar{W}

η

\bar{P}

and

\bar{E}

) optimization. The color on the Pareto frontier denotes the size of

\bar{E}

. To facilitate the observation of the changing relationships among the objectives, the pure red projection indicates the changing relationship between

\bar{W}

and

η

. The pure green projection shows the changing relationship between

\bar{W}

and

\bar{P}

, and the pure blue projection indicates the changing relationship between

η

and

\bar{P}

. It is easy to know that

\bar{W}

and

η

\bar{W}

and

\bar{P}

η

and

\bar{P}

are all parabolic-like relationships with the opening downward. To analyze the influence of the corresponding optimization variables (

{(u_{H})}_{opt}

{(u_{H 1})}_{opt}

{(u_{L})}_{opt}

and

π_{opt}

) on cycle performance, the distributions of

{(u_{H})}_{opt}

{(u_{H 1})}_{opt}

{(u_{L})}_{opt}

and

π_{opt}

within the Pareto frontier’s value range are shown in Figure 18, Figure 19, Figure 20 and Figure 21. As shown in Figure 18, the value range of

{(u_{H})}_{opt}

is 0–1, but its distribution is between 0.167 and 0.272. As

{(u_{H})}_{opt}

increases,

\bar{W}

\bar{P}

, and

\bar{E}

gradually increase, but

η

gradually decreases. As shown in Figure 19, the value range of

{(u_{H 1})}_{opt}

is 0–1, but its distribution is between 0.151 and 0.181. As

{(u_{H 1})}_{opt}

increases,

\bar{W}

\bar{P}

, and

\bar{E}

gradually decrease, but the changing trend of

η

is not apparent. As shown in Figure 20, the value range of

{(u_{L})}_{opt}

is 0–1, but its distribution is between 0.568 and 0.662. As

{(u_{L})}_{opt}

increases,

\bar{W}

\bar{P}

, and

\bar{E}

gradually decrease, but the changing trend of

η

is not apparent. As shown in Figure 21, the value range of

π_{opt}

is 2–50, but its distribution is between 9.692 and 24.426. As

π_{opt}

increases,

\bar{W}

gradually decreases,

η

gradually increases, and

\bar{P}

and

\bar{E}

rise and then reduce.

The Pareto frontier includes a series of non-inferior solutions, so the appropriate solution must be chosen according to the actual situation. The results of the triple- and double-objective optimizations are further discussed to compare the results of multi-objective optimizations more comprehensively. The comparison of the optimal schemes gotten by single- and double-, triple-, and quadruple-objective optimizations are listed in Table 2. The deviation index (D) is applied to represent the proximity between the optimal scheme and the positive ideal point. The appropriate optimal schemes are chosen by using the three methods. For the quadruple-objective optimization,

\bar{W}

η

\bar{P}

, and

\bar{E}

corresponding to the positive ideal point are the maximum of the single-objective optimization. It indicates that the Pareto frontier includes all single-objective optimization results. The D obtained by the Shannon Entropy method is significantly smaller than that obtained by the LINMAP and TOPSIS methods. Simultaneously, it can be found that the D obtained by the Shannon Entropy method is the same as that with

\bar{E}

as the OPO. For the triple-objective optimization, the triple-objective (

\bar{W}

η

and

\bar{E}

) optimization D obtained by the LINMAP or TOPSIS method is the smallest. For the double-objective optimization, the double-objective (

\bar{W}

and

\bar{P}

) optimization D obtained by the LINMAP method is the smallest. For the single-objective optimization, the D corresponding to

{\bar{E}}_{\max}

is the smallest. For single- and double-, triple-, and quadruple-objective optimizations, the double-objective (

\bar{W}

and

\bar{P}

) optimization D obtained by the LINMAP method is the smallest.

5. Conclusions

Based on FTT, an improved irreversible closed modified simple BCY model with one IHP and coupled to VTHRs is established and optimized with four performance indicators as OPOs, respectively. The optimization results are compared, and the influences of compressor and turbine efficiencies on optimization results are analyzed. Finally, the cycle is optimized, and the corresponding Pareto frontier is gained by adopting the NSGA-II algorithm. Based on three different methods, the optimal scheme is gotten from the Pareto frontier. The results obtained in this paper reveal the original results in Refs. [10,11,12], which were the initial work of the FTT theory. The main results are summarized:

For the single-objective analyses and optimizations, performance indicators all rise as $η_{c}$ and $η_{t}$ rise. The influences of $η_{t}$ on four performance indicators are greater than those of $η_{c}$ . $\bar{W}$ of the models in this paper increase and then decrease as $π$ increases in both cases; that is, the qualitative law is the same. However, there is an apparent quantitative difference between the two points. In practice, the difference between $τ_{H 1}$ and $τ_{H 3}$ should be controlled and not be too large. $\bar{P}$ and $\bar{E}$ are the trade-offs between $\bar{W}$ and $η$ .
For single- and double-, triple-, and quadruple-objective optimizations, the Pareto frontier includes a series of non-inferior solutions. The appropriate solution could be chosen according to the actual situation. By comparison, it is found that the double-objective ( $\bar{W}$ and $\bar{P}$ ) optimization D obtained by the LINMAP method is the smallest.
The optimization results gained in this paper could offer theoretical guidelines for the optimal designs of the gas turbine plants. In the next step, the improved closed intercooling regenerated modified BCY model with one IHP will be optimized with real gas as the working fluid, and the internal friction-based pressure drops during heating and cooling processes and other processes, as well as the heat leakage losses between the heat source and the environment, will be taken into account.

Author Contributions

Conceptualization: L.C. and H.F.; funding acquisition: L.C.; methodology: C.T.; software: C.T. and Y.G.; validation: L.C. and Y.G; writing—original draft: C.T. and H.F.; writing—review and editing: L.C. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 51779262).

Acknowledgments

The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

$a$ , $x$ , $y$	Intermediate variables
$C$	Thermal capacity rate (kW/K)
$C_{p}$	Specific heat at constant pressure (kJ/(kg·K))
$E$	Effectiveness of heat exchanger or ecological function (kW)
$\bar{E}$	Dimensionless ecological function
$k$	Specific heat ratio
$M$	Mach number
$N$	Number of the heat transfer unit
$\dot{Q}$	Heat absorbing rate or heat releasing rate (kW)
$\bar{P}$	Dimensionless power density
$T$	Temperature (K)
$U$	Heat conductance (kW/K)
$u$	Heat conductance distribution
$\bar{W}$	Dimensionless power output
Greek symbols
$η$	Efficiency
$π$	Pressure ratio
$τ$	Temperature ratio
Subscripts
$H$	Hot-side heat exchanger
$L$	Cold-side heat exchanger
$w f$	Working fluid
$1, 2, 3, 4, 5, 2 s, 5 s$	State points

Abbreviations

Brayton cycle	BCY
CCC	Convergent combustion chamber
CTHR	Constant-temperature heat reservoir
FPDT	Finite Physical Dimensions Thermodynamics
FTT	Finite time thermodynamics
HCD	Heat conductance distribution
IHP	Isothermal heating process
OPO	Optimization objective
RCC	Regular combustion chamber
VTHR	Variable-temperature heat reservoir

References

Wood, W.A. On the role of the harmonic mean isentropic exponent in the analysis of the closed-cycle gas turbine. Proc. Inst. Mech. Eng. Part. A J. Power Energy 1991, 205, 287–291. [Google Scholar] [CrossRef]
Cheng, K.L.; Qin, J.; Sun, H.C.; Li, H.; He, S.; Zhang, S.L.; Bao, W. Power optimization and comparison between simple recuperated and recompressing supercritical carbon dioxide Closed-Brayton-Cycle with finite cold source on hypersonic vehicles. Energy 2019, 181, 1189–1201. [Google Scholar] [CrossRef]
Hu, H.M.; Jiang, Y.Y.; Guo, C.H.; Liang, S.Q. Thermodynamic and exergy analysis of a S-CO2 Brayton cycle with various of cooling modes. Energy Convers. Manag. 2020, 220, 113110. [Google Scholar] [CrossRef]
Liu, H.Q.; Chi, Z.R.; Zang, S.S. Optimization of a closed Brayton cycle for space power systems. Appl. Therm. Eng. 2020, 179, 115611. [Google Scholar] [CrossRef]
Vecchiarelli, J.; Kawall, J.G.; Wallace, J.S. Analysis of a concept for increasing the efficiency of a Brayton cycle via isothermal heat addition. Int. J. Energy Res. 1997, 21, 113–127. [Google Scholar] [CrossRef]
Göktun, S.; Yavuz, H. Thermal efficiency of a regenerative Brayton cycle with isothermal heat addition. Energy Convers. Manag. 1999, 40, 1259–1266. [Google Scholar] [CrossRef]
Erbay, L.B.; Göktun, S.; Yavuz, H. Optimal design of the regenerative gas turbine engine with isothermal heat addition. Appl. Energy 2001, 68, 249–264. [Google Scholar] [CrossRef]
Jubeh, N.M. Exergy analysis and second law efficiency of a regenerative Brayton cycle with isothermal heat addition. Entropy 2005, 7, 172–187. [Google Scholar] [CrossRef] [Green Version]
El-Maksound, R.M.A. Binary Brayton cycle with two isothermal processes. Energy Convers. Manag. 2013, 73, 303–308. [Google Scholar] [CrossRef]
Moutier, J. Éléments de Thermodynamique; Gautier-Villars: Paris, France, 1872. [Google Scholar]
Novikov, I.I. The efficiency of atomic power stations (A review). J. Nucl. Energy 1957, 7, 125–128. [Google Scholar] [CrossRef]
Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 1975, 43, 22–24. [Google Scholar] [CrossRef]
Andresen, B. Finite-Time Thermodynamics; Physics Laboratory II; University of Copenhagen: Copenhagen, Denmark, 1983. [Google Scholar]
Grazzini, G. Work from irreversible heat engines. Energy 1991, 16, 747–755. [Google Scholar] [CrossRef]
Bejan, A. Entropy Generation Minimization; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
Chen, L.G.; Wu, C.; Sun, F.R. Finite time thermodynamic optimization or entropy generation minimization of energy systems. J. Non-Equilib. Thermodyn. 1999, 24, 327–359. [Google Scholar] [CrossRef]
Andresen, B. Current trends in finite-time thermodynamics. Angew. Chem. Int. Ed. 2011, 50, 2690–2704. [Google Scholar] [CrossRef] [PubMed]
Shittu, S.; Li, G.Q.; Zhao, X.D.; Ma, X.L. Review of thermoelectric geometry and structure optimization for performance enhancement. Appl. Energy 2020, 268, 115075. [Google Scholar] [CrossRef]
Berry, R.S.; Salamon, P.; Andresen, B. How it all began. Entropy 2020, 22, 908. [Google Scholar] [CrossRef] [PubMed]
Hoffman, K.H.; Burzler, J.; Fischer, A.; Schaller, M.; Schubert, S. Optimal process paths for endoreversible systems. J. Non-Equilib. Thermodyn. 2003, 28, 233–268. [Google Scholar] [CrossRef]
Zaeva, M.A.; Tsirlin, A.M.; Didina, O.V. Finite time thermodynamics: Realizability domain of heat to work converters. J. Non-Equilib. Thermodyn. 2019, 44, 181–191. [Google Scholar] [CrossRef]
Masser, R.; Hoffmann, K.H. Endoreversible modeling of a hydraulic recuperation system. Entropy 2020, 22, 383. [Google Scholar] [CrossRef] [Green Version]
Kushner, A.; Lychagin, V.; Roop, M. Optimal thermodynamic processes for gases. Entropy 2020, 22, 448. [Google Scholar] [CrossRef]
De Vos, A. Endoreversible models for the thermodynamics of computing. Entropy 2020, 22, 660. [Google Scholar] [CrossRef] [PubMed]
Masser, R.; Khodja, A.; Scheunert, M.; Schwalbe, K.; Fischer, A.; Paul, R.; Hoffmann, K.H. Optimized piston motion for an alpha-type Stirling engine. Entropy 2020, 22, 700. [Google Scholar] [CrossRef] [PubMed]
Chen, L.G.; Ma, K.; Ge, Y.L.; Feng, H.J. Re-optimization of expansion work of a heated working fluid with generalized radiative heat transfer law. Entropy 2020, 22, 720. [Google Scholar] [CrossRef] [PubMed]
Tsirlin, A.; Gagarina, L. Finite-time thermodynamics in economics. Entropy 2020, 22, 891. [Google Scholar] [CrossRef] [PubMed]
Tsirlin, A.; Sukin, I. Averaged optimization and finite-time thermodynamics. Entropy 2020, 22, 912. [Google Scholar] [CrossRef]
Muschik, W.; Hoffmann, K.H. Modeling, simulation, and reconstruction of 2-reservoir heat-to-power processes in finite-time thermodynamics. Entropy 2020, 22, 997. [Google Scholar] [CrossRef] [PubMed]
Insinga, A.R. The quantum friction and optimal finite-time performance of the quantum Otto cycle. Entropy 2020, 22, 1060. [Google Scholar] [CrossRef]
Schön, J.C. Optimal control of hydrogen atom-like systems as thermodynamic engines in finite time. Entropy 2020, 22, 1066. [Google Scholar] [CrossRef]
Andresen, B.; Essex, C. Thermodynamics at very long time and space scales. Entropy 2020, 22, 1090. [Google Scholar] [CrossRef]
Chen, L.G.; Ma, K.; Feng, H.J.; Ge, Y.L. Optimal configuration of a gas expansion process in a piston-type cylinder with generalized convective heat transfer law. Energies 2020, 13, 3229. [Google Scholar] [CrossRef]
Scheunert, M.; Masser, R.; Khodja, A.; Paul, R.; Schwalbe, K.; Fischer, A.; Hoffmann, K.H. Power-optimized sinusoidal piston motion and its performance gain for an Alpha-type Stirling engine with limited regeneration. Energies 2020, 13, 4564. [Google Scholar] [CrossRef]
Boikov, S.Y.; Andresen, B.; Akhremenkov, A.A.; Tsirlin, A.M. Evaluation of irreversibility and optimal organization of an integrated multi-stream heat exchange system. J. Non-Equilib. Thermodyn. 2020, 45, 155–171. [Google Scholar] [CrossRef]
Chen, L.G.; Feng, H.J.; Ge, Y.L. Maximum energy output chemical pump configuration with an infinite-low- and a finite-high-chemical potential mass reservoirs. Energy Convers. Manag. 2020, 223, 113261. [Google Scholar] [CrossRef]
Hoffmann, K.H.; Burzler, J.M.; Schubert, S. Endoreversible thermodynamics. J. Non-Equilib. Thermodyn. 1997, 22, 311–355. [Google Scholar]
Wagner, K.; Hoffmann, K.H. Endoreversible modeling of a PEM fuel cell. J. Non-Equilib. Thermodyn. 2015, 40, 283–294. [Google Scholar] [CrossRef]
Muschik, W. Concepts of phenominological irreversible quantum thermodynamics I: Closed undecomposed Schottky systems in semi-classical description. J. Non-Equilib. Thermodyn. 2019, 44, 1–13. [Google Scholar] [CrossRef] [Green Version]
Ponmurugan, M. Attainability of maximum work and the reversible efficiency of minimally nonlinear irreversible heat engines. J. Non-Equilib. Thermodyn. 2019, 44, 143–153. [Google Scholar] [CrossRef] [Green Version]
Raman, R.; Kumar, N. Performance analysis of Diesel cycle under efficient power density condition with variable specific heat of working fluid. J. Non-Equilib. Thermodyn. 2019, 44, 405–416. [Google Scholar] [CrossRef]
Schwalbe, K.; Hoffmann, K.H. Stochastic Novikov engine with Fourier heat transport. J. Non-Equilib. Thermodyn. 2019, 44, 417–424. [Google Scholar] [CrossRef]
Morisaki, T.; Ikegami, Y. Maximum power of a multistage Rankine cycle in low-grade thermal energy conversion. Appl. Thermal Eng. 2014, 69, 78–85. [Google Scholar] [CrossRef]
Yasunaga, T.; Ikegami, Y. Application of finite time thermodynamics for evaluation method of heat engines. Energy Proc. 2017, 129, 995–1001. [Google Scholar] [CrossRef]
Yasunaga, T.; Fontaine, K.; Morisaki, T.; Ikegami, Y. Performance evaluation of heat exchangers for application to ocean thermal energy conversion system. Ocean Thermal Energy Convers. 2017, 22, 65–75. [Google Scholar]
Yasunaga, T.; Koyama, N.; Noguchi, T.; Morisaki, T.; Ikegami, Y. Thermodynamical optimum heat source mean velocity in heat exchangers on OTEC. In Proceedings of the Grand Renewable Energy 2018, Yokohama, Japan, 17–22 June 2018. [Google Scholar]
Yasunaga, T.; Noguchi, T.; Morisaki, T.; Ikegami, Y. Basic heat exchanger performance evaluation method on OTEC. J. Mar. Sci. Eng. 2018, 6, 32. [Google Scholar] [CrossRef] [Green Version]
Fontaine, K.; Yasunaga, T.; Ikegami, Y. OTEC maximum net power output using Carnot cycle and application to simplify heat exchanger selection. Entropy 2019, 21, 1143. [Google Scholar] [CrossRef] [Green Version]
Yasunaga, T.; Ikegami, Y. Finite-time thermodynamic model for evaluating heat engines in ocean thermal energy conversion. Entropy 2020, 22, 211. [Google Scholar] [CrossRef] [Green Version]
Shittu, S.; Li, G.Q.; Zhao, X.D.; Ma, X.L.; Akhlaghi, Y.G.; Fan, Y. Comprehensive study and optimization of concentrated photovoltaic-thermoelectric considering all contact resistances. Energy Convers. Manag. 2020, 205, 112422. [Google Scholar] [CrossRef]
Feidt, M. Carnot cycle and heat engine: Fundamentals and applications. Entropy 2020, 22, 348. [Google Scholar] [CrossRef] [Green Version]
Feidt, M.; Costea, M. Effect of machine entropy production on the optimal performance of a refrigerator. Entropy 2020, 22, 913. [Google Scholar] [CrossRef] [PubMed]
Ma, Y.H. Effect of finite-size heat source’s heat capacity on the efficiency of heat engine. Entropy 2020, 22, 1002. [Google Scholar] [CrossRef] [PubMed]
Rogolino, P.; Cimmelli, V.A. Thermoelectric efficiency of Silicon–Germanium alloys in finite-time thermodynamics. Entropy 2020, 22, 1116. [Google Scholar] [CrossRef] [PubMed]
Dann, R.; Kosloff, R.; Salamon, P. Quantum finite time thermodynamics: Insight from a single qubit engine. Entropy 2020, 22, 1255. [Google Scholar] [CrossRef] [PubMed]
Liu, X.W.; Chen, L.G.; Ge, Y.L.; Feng, H.J.; Wu, F.; Lorenzini, G. Exergy-based ecological optimization of an irreversible quantum Carnot heat pump with spin-1/2 systems. J. Non-Equilib. Thermodyn. 2021, 46, 61–76. [Google Scholar] [CrossRef]
Guo, H.; Xu, Y.J.; Zhang, X.J.; Zhu, Y.L.; Chen, H.S. Finite-time thermodynamics modeling and analysis on compressed air energy storage systems with thermal storage. Renew. Sustain. Energy Rev. 2021, 138, 110656. [Google Scholar] [CrossRef]
Smith, Z.; Pal, P.S.; Deffner, S. Endoreversible Otto engines at maximal power. J. Non-Equilib. Thermodyn. 2020, 45, 305–310. [Google Scholar] [CrossRef]
Chen, L.G.; Shen, J.F.; Ge, Y.L.; Wu, Z.X.; Wang, W.H.; Zhu, F.L.; Feng, H.J. Power and efficiency optimization of open Maisotsenko-Brayton cycle and performance comparison with traditional open regenerated Brayton cycle. Energy Convers. Manag. 2020, 217, 113001. [Google Scholar] [CrossRef]
Liu, H.T.; Zhai, R.R.; Patchigolla, K.; Turner, P.; Yang, Y.P. Analysis of integration method in multi-heat-source power generation systems based on finite-time thermodynamics. Energy Convers. Manag. 2020, 220, 113069. [Google Scholar] [CrossRef]
Feng, H.J.; Qin, W.X.; Chen, L.G.; Cai, C.G.; Ge, Y.L.; Xia, S.J. Power output, thermal efficiency and exergy-based ecological performance optimizations of an irreversible KCS-34 coupled to variable temperature heat reservoirs. Energy Convers. Manag. 2020, 205, 112424. [Google Scholar] [CrossRef]
Feng, J.S.; Gao, G.T.; Dabwan, Y.N.; Pei, G.; Dong, H. Thermal performance evaluation of subcritical organic Rankine cycle for waste heat recovery from sinter annular cooler. J. Iron. Steel Res. Int. 2020, 27, 248–258. [Google Scholar] [CrossRef]
Wu, Z.X.; Feng, H.J.; Chen, L.G.; Tang, W.; Shi, J.C.; Ge, Y.L. Constructal thermodynamic optimization for ocean thermal energy conversion system with dual-pressure organic Rankine cycle. Energy Convers. Manag. 2020, 210, 112727. [Google Scholar] [CrossRef]
Qiu, S.S.; Ding, Z.M.; Chen, L.G. Performance evaluation and parametric optimum design of irreversible thermionic generators based on van der Waals heterostructures. Energy Convers. Manag. 2020, 225, 113360. [Google Scholar] [CrossRef]
Miller, H.J.D.; Mehboudi, M. Geometry of work fluctuations versus efficiency in microscopic thermal machines. Phys. Rev. Lett. 2020, 125, 260602. [Google Scholar] [CrossRef]
Gonzalez-Ayala, J.; Roco, J.M.M.; Medina, A.; Calvo Hernández, A. Optimization, stability, and entropy in endoreversible heat engines. Entropy 2020, 22, 1323. [Google Scholar] [CrossRef]
Kong, R.; Chen, L.G.; Xia, S.J.; Li, P.L.; Ge, Y.L. Minimizing entropy generation rate in hydrogen iodide decomposition reactor heated by high-temperature helium. Entropy 2021, 23, 82. [Google Scholar] [CrossRef]
Albatati, F.; Attar, A. Analytical and experimental study of thermoelectric generator (TEG) system for automotive exhaust waste heat recovery. Energies 2021, 14, 204. [Google Scholar] [CrossRef]
Feng, H.J.; Wu, Z.X.; Chen, L.G.; Ge, Y.L. Constructal thermodynamic optimization for dual-pressure organic Rankine cycle in waste heat utilization system. Energy Convers. Manag. 2021, 227, 113585. [Google Scholar] [CrossRef]
Garmejani, H.A.; Hossainpou, S.H. Single and multi-objective optimization of a TEG system for optimum power, cost and second law efficiency using genetic algorithm. Energy Convers. Manag. 2021, 228, 113658. [Google Scholar] [CrossRef]
Ge, Y.L.; Chen, L.G.; Feng, H.J. Ecological optimization of an irreversible Diesel cycle. Eur. Phys. J. Plus 2021, 136, 198. [Google Scholar] [CrossRef]
Chen, L.G.; Meng, F.K.; Ge, Y.L.; Feng, H.J.; Xia, S.J. Performance optimization of a class of combined thermoelectric heating devices. Sci. China Technol. Sci. 2020, 63, 2640–2648. [Google Scholar] [CrossRef]
Sahin, B.; Kodal, A.; Yavuz, H. Efficiency of a Joule-Brayton engine at maximum power density. J. Phys. D Appl. Phys. 1995, 28, 1309–1313. [Google Scholar] [CrossRef]
Sahin, B.; Kodal, A.; Yavuz, H. Maximum power density analysis of an endoreversible Carnot heat engine. Energy 1996, 21, 1219–1225. [Google Scholar] [CrossRef]
Chen, L.G.; Zheng, J.L.; Sun, F.R.; Wu, C. Optimum distribution of heat exchanger inventory for power density optimization of an endoreversible closed Brayton cycle. J. Phys. D Appl. Phys. 2001, 34, 422–427. [Google Scholar] [CrossRef]
Chen, L.G.; Zheng, J.L.; Sun, F.R.; Wu, C. Power density optimization for an irreversible closed Brayton cycle. Open Syst. Inf. Dyn. 2001, 8, 241–260. [Google Scholar] [CrossRef]
Chen, L.G.; Zheng, J.L.; Sun, F.R.; Wu, C. Performance comparison of an endoreversible closed variable-temperature heat reservoir Brayton cycle under maximum power density and maximum power conditions. Energy Convers. Manag. 2002, 43, 33–43. [Google Scholar] [CrossRef]
Chen, L.G.; Zheng, J.L.; Sun, F.R.; Wu, C. Performance comparison of an irreversible closed variable-temperature heat reservoir Brayton cycle under maximum power density and maximum power conditions. Proc. Inst. Mech. Eng. Part. A J. Power Energy 2005, 219, 559–566. [Google Scholar] [CrossRef]
Gonca, G. Thermodynamic analysis and performance maps for the irreversible Dual-Atkinson cycle engine (DACE) with considerations of temperature-dependent specific heats, heat transfer and friction losses. Energy Convers. Manag. 2016, 111, 205–216. [Google Scholar] [CrossRef]
Gonca, G.; Bahri Sahin, B.; Cakir, M. Performance assessment of a modified power generating cycle based on effective ecological power density and performance coefficient. Int. J. Exergy 2020, 33, 153–164. [Google Scholar] [CrossRef]
Karakurt, A.S.; Bashan, V.; Ust, Y. Comparative maximum power density analysis of a supercritical CO₂ Brayton power cycle. J. Therm. Eng. 2020, 6, 50–57. [Google Scholar] [CrossRef]
Angulo-Brown, F. An ecological optimization criterion for finite-time heat engines. J. Appl. Phys. 1991, 69, 7465–7469. [Google Scholar] [CrossRef]
Yan, Z.J. Comment on “ecological optimization criterion for finite-time heat engines”. Eur. J. Appl. Physiol. 1993, 73, 3583. [Google Scholar]
Cheng, C.Y.; Chen, C.K. Ecological optimization of an endoreversible Brayton cycle. Energy Convers. Manag. 1998, 39, 33–44. [Google Scholar] [CrossRef]
Ma, Z.S.; Chen, Y.; Wu, J.H. Ecological optimization for a combined diesel-organic Rankine cycle. AIP Adv. 2019, 9, 015320. [Google Scholar] [CrossRef] [Green Version]
Ahmadi, M.H.; Pourkiaei, S.M.; Ghazvini, M.; Pourfayaz, F. Thermodynamic assessment and optimization of performance of irreversible Atkinson cycle. Iran. J. Chem. Chem. Eng. 2020, 39, 267–280. [Google Scholar]
Levario-Medina, S.; Valencia-Ortega, G.; Barranco-Jimenez, M.A. Energetic optimization considering a generalization of the ecological criterion in traditional simple-cycle and combined cycle power plants. J. Non-Equilib. Thermodyn. 2020, 45, 269–290. [Google Scholar] [CrossRef]
Wu, H.; Ge, Y.L.; Chen, L.G.; Feng, H.J. Power, efficiency, ecological function and ecological coefficient of performance optimizations of an irreversible Diesel cycle based on finite piston speed. Energy 2021, 216, 119235. [Google Scholar] [CrossRef]
Kaushik, S.C.; Tyagi, S.K.; Singhal, M.K. Parametric study of an irreversible regenerative Brayton cycle with isothermal heat addition. Energy Convers. Manag. 2003, 44, 2013–2025. [Google Scholar] [CrossRef]
Tyagi, S.K.; Kaushik, S.C.; Tiwari, V. Ecological optimization and parametric study of an irreversible regenerative modified Brayton cycle with isothermal heat addition. Entropy 2003, 5, 377–390. [Google Scholar] [CrossRef] [Green Version]
Tyagi, S.K.; Chen, J. Performance evaluation of an irreversible regenerative modified Brayton heat engine based on the thermoeconomic criterion. Int. J. Power Energy Syst. 2006, 26, 66–74. [Google Scholar] [CrossRef]
Kumar, R.; Kaushik, S.C.; Kumar, R. Power optimization of an irreversible regenerative Brayton cycle with isothermal heat addition. J. Therm. Eng. 2015, 1, 279–286. [Google Scholar] [CrossRef]
Tyagi, S.K.; Chen, J.; Kaushik, S.C. Optimum criteria based on the ecological function of an irreversible intercooled regenerative modified Brayton cycle. Int. J. Exergy 2005, 2, 90–107. [Google Scholar] [CrossRef]
Tyagi, S.K.; Wang, S.; Kaushik, S.C. Irreversible modified complex Brayton cycle under maximum economic condition. Indian J. Pure Appl. Phys. 2006, 44, 592–601. [Google Scholar]
Tyagi, S.K.; Chen, J.; Kaushik, S.C.; Wu, C. Effects of intercooling on the performance of an irreversible regenerative modified Brayton cycle. Int. J. Power Energy Syst. 2007, 27, 256–264. [Google Scholar] [CrossRef]
Tyagi, S.K.; Wang, S.; Park, S.R. Performance criteria on different pressure ratios of an irreversible modified complex Brayton cycle. Indian J. Pure Appl. Phys. 2008, 46, 565–574. [Google Scholar]
Wang, J.H.; Chen, L.G.; Ge, Y.L.; Sun, F.R. Power and power density analyzes of an endoreversible modified variable-temperature reservoir Brayton cycle with isothermal heat addition. Int. J. Low-Carbon Technol. 2016, 11, 42–53. [Google Scholar] [CrossRef] [Green Version]
Wang, J.H.; Chen, L.G.; Ge, Y.L.; Sun, F.R. Ecological performance analysis of an endoreversible modified Brayton cycle. Int. J. Sustain. Energy 2014, 33, 619–634. [Google Scholar] [CrossRef]
Tang, C.Q.; Feng, H.J.; Chen, L.G.; Wang, W.H. Power density analysis and multi-objective optimization for a modified endoreversible simple closed Brayton cycle with one isothermal heating process. Energy Rep. 2020, 6, 1648–1657. [Google Scholar] [CrossRef]
Arora, R.; Kaushik, S.C.; Kumar, R.; Arora, R. Soft computing based multi-objective optimization of Brayton cycle power plant with isothermal heat addition using evolutionary algorithm and decision making. Appl. Soft Comput. 2016, 46, 267–283. [Google Scholar] [CrossRef]
Arora, R.; Arora, R. Thermodynamic optimization of an irreversible regenerated Brayton heat engine using modified ecological criteria. J. Therm. Eng. 2020, 6, 28–42. [Google Scholar] [CrossRef]
Chen, L.G.; Tang, C.Q.; Feng, H.J.; Ge, Y.L. Power, efficiency, power density and ecological function optimizations for an irreversible modified closed variable-temperature reservoir regenerative Brayton cycle with one isothermal heating process. Energies 2020, 13, 5133. [Google Scholar] [CrossRef]
Qi, W.; Wang, W.H.; Chen, L.G. Power and efficiency performance analyses for a closed endoreversible binary Brayton cycle with two isothermal processes. Therm. Sci. Eng. Prog. 2018, 7, 131–137. [Google Scholar] [CrossRef]
Tang, C.Q.; Chen, L.G.; Feng, H.J.; Wang, W.H.; Ge, Y.L. Power optimization of a closed binary Brayton cycle with isothermal heating processes and coupled to variable-temperature reservoirs. Energies 2020, 13, 3212. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Dehghani, S.; Mohammadi, A.H.; Feidt, M.; Barranco-Jimenez, M.A. Optimal design of a solar driven heat engine based on thermal and thermo-economic criteria. Energy Convers. Manag. 2013, 75, 635–642. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Mohammadi, A.H.; Dehghani, S.; Barranco-Jimenez, M.A. Multi-objective thermodynamic-based optimization of output power of Solar Dish-Stirling engine by implementing an evolutionary algorithm. Energy Convers. Manag. 2013, 75, 438–445. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Ahmadi, M.A.; Mohammadi, A.H.; Feidt, M.; Pourkiaei, S.M. Multi-objective optimization of an irreversible Stirling cryogenic refrigerator cycle. Energy Convers. Manag. 2014, 82, 351–360. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Ahmadi, M.A.; Mehrpooya, M.; Hosseinzade, H.; Feidt, M. Thermodynamic and thermo-economic analysis and optimization of performance of irreversible four- temperature-level absorption refrigeration. Energy Convers. Manag. 2014, 88, 1051–1059. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Ahmadi, M.A. Thermodynamic analysis and optimization of an irreversible Ericsson cryogenic refrigerator cycle. Energy Convers. Manag. 2015, 89, 147–155. [Google Scholar] [CrossRef]
Jokar, M.A.; Ahmadi, M.H.; Sharifpur, M.; Meyer, J.P.; Pourfayaz, F.; Ming, T.Z. Thermodynamic evaluation and multi-objective optimization of molten carbonate fuel cell-supercritical CO2 Brayton cycle hybrid system. Energy Convers. Manag. 2017, 153, 538–556. [Google Scholar] [CrossRef]
Han, Z.H.; Mei, Z.K.; Li, P. Multi-objective optimization and sensitivity analysis of an organic Rankine cycle coupled with a one-dimensional radial-inflow turbine efficiency prediction model. Energy Convers. Manag. 2018, 166, 37–47. [Google Scholar] [CrossRef]
Ghasemkhani, A.; Farahat, S.; Naserian, M.M. Multi-objective optimization and decision making of endoreversible combined cycles with consideration of different heat exchangers by finite time thermodynamics. Energy Convers. Manag. 2018, 171, 1052–1062. [Google Scholar] [CrossRef]
Ahmadi, M.H.; Jokar, M.A.; Ming, T.Z.; Feidt, M.; Pourfayaz, F.; Astaraei, F.R. Multi-objective performance optimization of irreversible molten carbonate fuel cell–Braysson heat engine and thermodynamic analysis with ecological objective approach. Energy 2018, 144, 707–722. [Google Scholar] [CrossRef]
Wang, M.; Jing, R.; Zhang, H.R.; Meng, C.; Li, N.; Zhao, Y.R. An innovative Organic Rankine Cycle (ORC) based Ocean Thermal Energy Conversion (OTEC) system with performance simulation and multi-objective optimization. Appl. Therm. Eng. 2018, 145, 743–754. [Google Scholar] [CrossRef]
Patela, V.K.; Raja, B.D. A comparative performance evaluation of the reversed Brayton cycle operated heat pump based on thermo-ecological criteria through many and multi-objective approaches. Energy Convers. Manag. 2019, 183, 252–265. [Google Scholar] [CrossRef]
Hu, S.Z.; Li, J.; Yang, F.B.; Yang, Z.; Duan, Y.Y. Multi-objective optimization of organic Rankine cycle using hydrofluorolefins (HFOs) based on different target preferences. Energy 2020, 203, 117848. [Google Scholar] [CrossRef]
Hu, S.Z.; Li, J.; Yang, F.B.; Yang, Z.; Duan, Y.Y. How to design organic Rankine cycle system under fluctuating ambient temperature: A multi-objective approach. Energy Convers. Manag. 2020, 224, 113331. [Google Scholar] [CrossRef]
Sun, M.; Xia, S.J.; Chen, L.G.; Wang, C.; Tang, C.Q. Minimum entropy generation rate and maximum yield optimization of sulfuric acid decomposition process using NSGA-II. Entropy 2020, 22, 1065. [Google Scholar] [CrossRef] [PubMed]
Sadeghi, S.; Ghandehariun, S.; Naterer, G.F. Exergoeconomic and multi-objective optimization of a solar thermochemical hydrogen production plant with heat recovery. Energy Convers. Manag. 2020, 225, 113441. [Google Scholar] [CrossRef]
Wu, Z.X.; Feng, H.J.; Chen, L.G.; Ge, Y.L. Performance optimization of a condenser in ocean thermal energy conversion (OTEC) system based on constructal theory and multi-objective genetic algorithm. Entropy 2020, 22, 641. [Google Scholar] [CrossRef]
Ghorani, M.M.; Haghighi, M.H.S.; Riasi, A. Entropy generation minimization of a pump running in reverse mode based on surrogate models and NSGA-II. Int. Commun. Heat Mass Transfer 2020, 118, 104898. [Google Scholar] [CrossRef]
Wang, L.B.; Bu, X.B.; Li, H.S. Multi-objective optimization and off-design evaluation of organic Rankine cycle (ORC) for low-grade waste heat recovery. Energy 2020, 203, 117809. [Google Scholar] [CrossRef]
Herrera-Orozco, I.; Valencia-Ochoa, G.; Jorge Duarte-Forero, J. Exergo-environmental assessment and multi-objective optimization of waste heat recovery systems based on Organic Rankine cycle configurations. J. Clean. Prod. 2021, 288, 125679. [Google Scholar] [CrossRef]
Shi, S.S.; Ge, Y.L.; Chen, L.G.; Feng, F.J. Four objective optimization of irreversible Atkinson cycle based on NSGA-II. Entropy 2020, 22, 1150. [Google Scholar] [CrossRef]
Tang, W.; Feng, H.J.; Chen, L.G.; Xie, Z.J.; Shi, J.C. Constructal design for a boiler economizer. Energy 2021, 223, 120013. [Google Scholar] [CrossRef]
Chen, L.G.; Zheng, J.L.; Sun, F.R.; Wu, C. Power density optimization for an irreversible regenerated closed Brayton cycle. Phys. Scripta 2001, 64, 184–191. [Google Scholar] [CrossRef]
Bejan, A. Entropy Generation through Heat and Fluid Flow; Wiley: New York, NY, USA, 1982. [Google Scholar]
Bejan, A. Theory of heat transfer-irreversible power plant. Int. J. Heat Mass Transfer 1988, 31, 1211–1219. [Google Scholar] [CrossRef]
Bejan, A. The equivalence of maximum power and minimum entropy generation rate in the optimization of power plants. J. Energy Res. Tech. 1996, 118, 98–101. [Google Scholar] [CrossRef]
Bejan, A. Models of power plants that generate minimum entropy while operating at maximum power. Am. J. Phys. 1996, 64, 1054–1059. [Google Scholar] [CrossRef]
Salamon, P.; Hoffmann, K.H.; Schubert, S.; Berry, R.S.; Andresen, B. What conditions make minimum entropy production equivalent to maximum power production? J. Non-Equilib. Thermodyn. 2001, 26, 73–83. [Google Scholar] [CrossRef] [Green Version]
Andresen, B.; Berry, R.S.; Nitzan, A.; Salamon, P. Thermodynamics in finite time: The step-Carnot cycle. Phys. Rev. A 1977, 15, 2086–2093. [Google Scholar] [CrossRef]
Orlov, V.N.; Rudenko, A.V. Optimal control in problems of extremal of irreversible thermodynamic processes. Avtomatika Telemekhanika 1985, 46, 549–577. [Google Scholar]
Lu, P.C. Thermodynamics with finite heat-transfer area or finite surface thermodynamics. Thermodynamics and the Design, Analysis, and Improvement of Energy Systems, ASME Adv. Energy Sys. Div. Pub. AES 1995, 35, 51–60. [Google Scholar]
Bejan, A. Entropy generation minimization: The new thermodynamics of finite size devices and finite time processes. J. Appl. Phys. 1996, 79, 1191–1218. [Google Scholar] [CrossRef] [Green Version]
Feidt, M. Thermodynamique et Optimisation Energetique des Systems et Procedes, 2nd ed.; Technique et Documentation, Lavoisier: Paris, France, 1996. (In French) [Google Scholar]
Dong, Y.; El-Bakkali, A.; Feidt, M.; Descombes, G.; Perilhon, C. Association of finite-dimension thermodynamics and a bond-graph approach for modeling an irreversible heat engine. Entropy 2012, 14, 1234–1258. [Google Scholar] [CrossRef] [Green Version]
Feidt, M. Thermodynamique Optimale en Dimensions Physiques Finies; Hermès: Paris, France, 2013. [Google Scholar]
Perescu, S.; Costea, M.; Feidt, M.; Ganea, I.; Boriaru, N. Advanced Thermodynamics of Irreversible Processes with Finite Speed and Finite Dimensions; Editura AGIR: Bucharest, Romania, 2015. [Google Scholar]
Feidt, M. Finite Physical Dimensions Optimal Thermodynamics 1. Fundamental; ISTE Press and Elsevier: London, UK, 2017. [Google Scholar]
Feidt, M. Finite Physical Dimensions Optimal Thermodynamics 2. Complex. Systems; ISTE Press and Elsevier: London, UK, 2018. [Google Scholar]
Blaise, M.; Feidt, M.; Maillet, D. Influence of the working fluid properties on optimized power of an irreversible finite dimensions Carnot engine. Energy Convers. Manag. 2018, 163, 444–456. [Google Scholar] [CrossRef]
Feidt, M.; Costea, M. From finite time to finite physical dimensions thermodynamics: The Carnot engine and Onsager’s relations revisited. J. Non-Equilib. Thermodyn. 2018, 43, 151–162. [Google Scholar] [CrossRef]
Dumitrascu, G.; Feidt, M.; Popescu, A.; Grigorean, S. Endoreversible trigeneration cycle design based on finite physical dimensions thermodynamics. Energies 2019, 12, 3165. [Google Scholar] [CrossRef] [Green Version]
Feidt, M.; Costea, M. Progress in Carnot and Chambadal modeling of thermomechnical engine by considering entropt and heat transfer entropy. Entropy 2019, 21, 1232. [Google Scholar] [CrossRef] [Green Version]
Feidt, M.; Costea, M.; Feidt, R.; Danel, Q.; Périlhon, C. New criteria to characterize the waste heat recovery. Energies 2020, 13, 789. [Google Scholar] [CrossRef] [Green Version]

Figure 1. Schematic diagram of the cycle.

Figure 2. Diagram of the cycle.

Figure 3. Diagrams of (a) irreversible simple BCY with an IHP and coupled to CTHRs; (b) endoreversible simple BCY with an IHP and coupled to VTHRs; (c) endoreversible simple BCY with an IHP and coupled to CTHRs; (d) simple irreversi-ble BCY coupled to VTHRs; (e) simple irreversible BCY coupled to CTHRs; (f) simple endoreversible BCY coupled to VTHRs; (g) simple endoreversible BCY coupled to CTHRs; (h) endoreversible Carnot cycle coupled to VTHRs; (i) endoreversible Carnot cycle coupled to CTHRs; (j) endoreversible Novikov cycle coupled to CTHRs.

Figure 4. Relationships of

\bar{W}

and

η

versus

π

with different

η_{t}

Figure 4. Relationships of

\bar{W}

and

η

versus

π

with different

η_{t}

Figure 5. Relationships of

\bar{P}

and

\bar{E}

versus

π

with different

η_{t}

Figure 5. Relationships of

\bar{P}

and

\bar{E}

versus

π

with different

η_{t}

Figure 6. Relationships of

π_{t}

and

v_{5} / v_{1}

versus

π

with different

η_{t}

Figure 6. Relationships of

π_{t}

and

v_{5} / v_{1}

versus

π

with different

η_{t}

Figure 7. Comparison of

\bar{W}

under the variable and constant

π

Figure 7. Comparison of

\bar{W}

under the variable and constant

π

Figure 8. Flowchart of HCD optimization.

Figure 9. Relationships of

{\bar{W}}_{η_{opt}}

and

η_{opt}

versus

π

Figure 9. Relationships of

{\bar{W}}_{η_{opt}}

and

η_{opt}

versus

π

Figure 10. Relationships of

{\bar{P}}_{η_{opt}}

and

{\bar{E}}_{η_{opt}}

versus

π

Figure 10. Relationships of

{\bar{P}}_{η_{opt}}

and

{\bar{E}}_{η_{opt}}

versus

π

Figure 11. Relationships of

{(π_{t})}_{η_{opt}}

and

{(v_{5} / v_{1})}_{η_{opt}}

versus

π

Figure 11. Relationships of

{(π_{t})}_{η_{opt}}

and

{(v_{5} / v_{1})}_{η_{opt}}

versus

π

Figure 12. Relationships of

{(u_{H})}_{η_{opt}}

{(u_{H 1})}_{η_{opt}}

and

{(u_{L})}_{η_{opt}}

versus

π

Figure 12. Relationships of

{(u_{H})}_{η_{opt}}

{(u_{H 1})}_{η_{opt}}

and

{(u_{L})}_{η_{opt}}

versus

π

Figure 13. Relationships of

η_{\max}

versus

τ_{H 1}

and

τ_{H 3}

Figure 13. Relationships of

η_{\max}

versus

τ_{H 1}

and

τ_{H 3}

Figure 14. Relationships of

\bar{W}

under various optimal performance indexes versus

η_{c}

and

η_{t}

Figure 14. Relationships of

\bar{W}

under various optimal performance indexes versus

η_{c}

and

η_{t}

Figure 15. Relationships of

π

under various optimal performance indexes versus

η_{c}

and

η_{t}

Figure 15. Relationships of

π

under various optimal performance indexes versus

η_{c}

and

η_{t}

Figure 16. Flowchart of NSGA-II algorithm.

Figure 17. Pareto frontier and optimal schemes corresponding to the four objectives (

\bar{W}

η

\bar{P}

and

\bar{E}

) optimization.

Figure 17. Pareto frontier and optimal schemes corresponding to the four objectives (

\bar{W}

η

\bar{P}

and

\bar{E}

) optimization.

Figure 18. Distribution of

{(u_{H})}_{opt}

within the value range in the Pareto frontier.

Figure 18. Distribution of

{(u_{H})}_{opt}

within the value range in the Pareto frontier.

Figure 19. Distribution of

{(u_{H 1})}_{opt}

within the value range in the Pareto frontier.

Figure 19. Distribution of

{(u_{H 1})}_{opt}

within the value range in the Pareto frontier.

Figure 20. Distribution of

{(u_{L})}_{opt}

within the value range in the Pareto frontier.

Figure 20. Distribution of

{(u_{L})}_{opt}

within the value range in the Pareto frontier.

Figure 21. Distribution of

π_{opt}

within the value range in the Pareto frontier.

Figure 21. Distribution of

π_{opt}

within the value range in the Pareto frontier.

Table 1. Parameter settings of “gamultiobj”.

Parameters	Values
Nvars	4
ParetoFraction	0.3
PopulationSize	300
Generations	500
CrossoverFraction	0.8

Table 2. Comparison of the optimal schemes gotten by the single- and double-, triple-, and quadruple-objective optimizations.

OPOs	Decision Methods	Optimization Variables				Performance Indicators				Isothermal Pressure Drop Ratio	Deviation Indexes
OPOs	Decision Methods	$u_{H}$	$u_{H 1}$	$u_{L}$	$π$	$\bar{W}$	$η$	$\bar{P}$	$\bar{E}$	$π_{t}$	$D$
$\bar{W}$ , $η$ , $\bar{P}$ , and $\bar{E}$	LINMAP	0.245	0.154	0.601	14.194	0.787	0.397	0.380	0.462	0.572	0.172
	TOPSIS	0.245	0.154	0.601	14.194	0.787	0.397	0.380	0.462	0.572	0.172
	Shannon Entropy	0.259	0.151	0.590	11.901	0.802	0.386	0.376	0.467	0.572	0.167
$\bar{W}$ , $η$ , and $\bar{P}$	LINMAP	0.230	0.167	0.603	14.261	0.787	0.398	0.381	0.461	0.557	0.170
	TOPSIS	0.231	0.167	0.602	14.115	0.788	0.397	0.380	0.462	0.557	0.168
	Shannon Entropy	0.246	0.167	0.587	12.008	0.803	0.386	0.377	0.466	0.559	0.165
$\bar{W}$ , $η$ , and $\bar{E}$	LINMAP	0.231	0.163	0.606	13.947	0.790	0.397	0.380	0.462	0.557	0.160
	TOPSIS	0.231	0.163	0.606	13.947	0.790	0.397	0.380	0.462	0.557	0.160
	Shannon Entropy	0.257	0.153	0.590	11.92	0.803	0.386	0.376	0.467	0.570	0.165
$\bar{W}$ , $\bar{P}$ and $\bar{E}$	LINMAP	0.252	0.177	0.571	13.339	0.793	0.393	0.380	0.463	0.566	0.162
	TOPSIS	0.252	0.177	0.571	13.339	0.793	0.393	0.380	0.463	0.566	0.162
	Shannon Entropy	0.259	0.151	0.590	11.906	0.802	0.386	0.376	0.467	0.572	0.167
$η$ , $\bar{P}$ and $\bar{E}$	LINMAP	0.241	0.170	0.589	17.016	0.761	0.406	0.380	0.444	0.575	0.319
	TOPSIS	0.241	0.170	0.589	17.016	0.761	0.406	0.380	0.444	0.575	0.319
	Shannon Entropy	0.245	0.169	0.585	16.674	0.764	0.405	0.381	0.447	0.577	0.297
$\bar{W}$ and $η$	LINMAP	0.230	0.169	0.601	14.293	0.787	0.398	0.381	0.461	0.557	0.170
	TOPSIS	0.230	0.169	0.601	14.293	0.787	0.398	0.381	0.461	0.557	0.170
	Shannon Entropy	0.248	0.168	0.585	12.061	0.802	0.387	0.377	0.466	0.560	0.162
$\bar{W}$ and $\bar{P}$	LINMAP	0.247	0.176	0.578	13.384	0.793	0.394	0.380	0.463	0.563	0.158
	TOPSIS	0.247	0.176	0.577	13.560	0.792	0.394	0.381	0.463	0.564	0.161
	Shannon Entropy	0.245	0.171	0.584	11.855	0.803	0.385	0.376	0.466	0.555	0.170
$\bar{W}$ and $\bar{E}$	LINMAP	0.258	0.154	0.589	11.765	0.803	0.385	0.376	0.467	0.570	0.169
	TOPSIS	0.258	0.154	0.589	11.765	0.803	0.385	0.376	0.467	0.570	0.169
	Shannon Entropy	0.259	0.152	0.589	11.902	0.802	0.386	0.376	0.467	0.572	0.167
$η$ and $\bar{P}$	LINMAP	0.232	0.192	0.576	16.452	0.765	0.405	0.381	0.446	0.562	0.295
	TOPSIS	0.235	0.193	0.572	16.156	0.768	0.404	0.381	0.447	0.563	0.279
	Shannon Entropy	0.241	0.196	0.563	15.603	0.772	0.402	0.381	0.450	0.564	0.255
$η$ and $\bar{E}$	LINMAP	0.237	0.1604	0.603	14.307	0.787	0.398	0.381	0.461	0.564	0.170
	TOPSIS	0.236	0.163	0.601	14.173	0.788	0.398	0.381	0.462	0.562	0.164
	Shannon Entropy	0.258	0.152	0.590	11.909	0.802	0.386	0.376	0.467	0.571	0.167
$\bar{P}$ and $\bar{E}$	LINMAP	0.257	0.166	0.578	13.483	0.792	0.394	0.380	0.464	0.572	0.160
	TOPSIS	0.257	0.165	0.578	13.386	0.793	0.393	0.380	0.464	0.572	0.161
	Shannon Entropy	0.258	0.154	0.588	12.054	0.802	0.387	0.377	0.467	0.571	0.161
$\bar{W}$		0.249	0.162	0.589	9.678	0.810	0.369	0.365	0.459	0.550	0.242
$η$		0.152	0.174	0.674	24.542	0.672	0.416	0.358	0.369	0.532	0.783
$\bar{P}$		0.251	0.183	0.567	15.149	0.777	0.400	0.382	0.454	0.571	0.225
$\bar{E}$		0.259	0.151	0.590	11.903	0.802	0.386	0.376	0.467	0.572	0.167
Positive ideal point		——	——	——	——	0.810	0.416	0.382	0.467	0.810	——
Negative ideal point		——	——	——	——	0.677	0.369	0.360	0.373	0.677	——

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Tang, C.; Chen, L.; Feng, H.; Ge, Y. Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle. Entropy 2021, 23, 282. https://doi.org/10.3390/e23030282

AMA Style

Tang C, Chen L, Feng H, Ge Y. Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle. Entropy. 2021; 23(3):282. https://doi.org/10.3390/e23030282

Chicago/Turabian Style

Tang, Chenqi, Lingen Chen, Huijun Feng, and Yanlin Ge. 2021. "Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle" Entropy 23, no. 3: 282. https://doi.org/10.3390/e23030282

APA Style

Tang, C., Chen, L., Feng, H., & Ge, Y. (2021). Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle. Entropy, 23(3), 282. https://doi.org/10.3390/e23030282

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle

Abstract

1. Introduction

2. Cycle Model and Performance Analytical Indicators

3. Analyses and Optimizations with Each Single Objective

3.1. Analyses of Each Single Objective

3.2. Performance Optimizations for Each Single Objective

3.2.1. Optimizations of Each Single Objective

3.2.2. Influences of Temperature Ratios on Optimization Results

3.2.3. Influences of the Compressor and the Turbine’s Irreversibilities on Optimization Results

4. Multi-Objective Optimization

4.1. Optimization Algorithm and Decision-Making Methods

4.2. Multi-Objective Optimization Results

5. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

Nomenclature

Abbreviations

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI