Power, Efficiency, Power Density and Ecological Function Optimization for an Irreversible Modified Closed Variable-Temperature Reservoir Regenerative Brayton Cycle with One Isothermal Heating Process

Abstract

One or more isothermal heating process was introduced to modify single and regenerative Brayton cycles by some scholars, which effectively improved the thermal efficiency and significantly reduced the emissions. To analyze and optimize the performance of this type of Brayton cycle, a regenerative modified Brayton cycle with an isothermal heating process is established in this paper based on finite time thermodynamics. The isothermal pressure drop ratio is variable. The irreversibilities of the compressor, turbine and all heat exchangers are considered in the cycle, and the heat reservoirs are variable-temperature ones. The function expressions of four performance indexes; that is, dimensionless power output, thermal efficiency, dimensionless power density and dimensionless ecological function are obtained. With the dimensionless power density as the optimization objective, the heat conductance distributions among all heat exchangers and the thermal capacitance rate matching among the working fluid and heat reservoir are optimized. Based on the NSGA-II algorithm, the cycle’s double-, triple- and quadruple-objective optimization are conducted with the total pressure ratio and the heat conductance distributions among heat exchangers as design variables. The optimal value is chosen from the Pareto frontier by applying the LINMAP, TOPSIS and Shannon entropy methods. The results show that when the pressure ratio in the compressor is less than 12.0, it is beneficial to add the regenerator to improve the cycle performance; when the pressure ratio is greater than 12.0, adding the regenerator will reduce the cycle performance. For single-objective optimization, the four performance indexes could be maximized under the optimal pressure ratios, respectively. When the pressure ratio is greater than 9.2, the cycle is simplified to a closed irreversible simple modified Brayton cycle with one isothermal heating process and coupled to variable-temperature heat reservoirs. Therefore, when the regenerator is used, the range of pressure ratio is limited, and a suitable pressure ratio should be selected. The triple objective (dimensionless power output, dimensionless power density and dimensionless ecological function) optimization’ deviation index gained by LINMAP or TOPSIS method is the smallest. The optimization results gained in this paper could offer some new pointers for the regenerative Brayton cycles’ optimal designs.

1. Introduction

Gas turbine plant (Brayton cycle (BCY)) has the characteristics of high safety, high efficiency and high energy density, and is widely used in the fields of thermal power generation, aviation propulsion, pipeline transportation, distributed energy and ship power plant, etc. However, NOx in gas turbine emission pollutants causes serious environmental pollution. The concentration of NOx in pollutants is closely related to the temperature of the combustion. For this reason, Vecchiarelli et al. [1] introduced an isothermal heating process (IHP) into the BCY, which effectively improved the thermal efficiency (TEF) and significantly reduced the emissions. The regenerative BCYs with IHP were studied by using the classical thermodynamics in [2,3,4]. El-Maksoud [5] added two IHPs in a binary BCY, and the cycle performance was significantly improved.

The temperature difference between the working fluid (WF) and the heat reservoir is not considered in the classical thermodynamic analysis. Compared to the classical thermodynamics, the finite size device and the finite time process of the finite rate heat exchange between the system and the environment are considered in the finite time thermodynamics (FTT) [6] and see the books by Berry et al. [7], Dincer et al. [8], Perescu et al. [9], Kaushik et al. [10] and Sieniutycz [11], as well as the review papers by Chen et al. [12], Andresen [13], Roach et al. [14] and Feidt et al. [15]. Berry et al. [7] comprehensively summarized the previous nearly 20 years’ achievements on the FTT and optimal control. Dincer et al. [8] studied all kinds of advanced multi-output thermodynamic cycles and found that compared with traditional thermodynamic cycles, the advanced cycles had higher sustainability and energy utilization efficiency. Perescu et al. [9] studied how to meet the electricity and hot water needed for the personal housing by designing suitable mirrors. By using the FTT, Kaushik et al. [10] studied Carnot, Rankine, Brayton, Stirling, Ericsson Cycles, etc. Sieniutycz [11] described the properties of complexity and complex thermo-economic systems. Chen et al. [12] discussed the performance optimization of the thermodynamic processes and devices under the constraint of finite time or finite size. Andresen [13] introduced the development of the FTT in the past ten years and focused on the methods that he considered to be the most promising. Roach et al. [14] discussed the application of the principles of the FTT and control thermodynamics to biological processes. Feidt et al. [15] suggested discussing the heat transfer entropy and the production of entropy to study the external or internal irreversibility of the converter. The FTT has been widely used in thermodynamic cycle researches, the power output [16,17,18], TEF [16,17,18], power density [19,20,21,22,23] and ecological function [24,25,26,27,28] are often targeted when applying FTT to analyze or optimize the cycle performances.

Kaushik et al. [29] applied FTT in the study of the regenerative BCY with an IHP and found that compared with the traditional BCY, the TEF of the new BCY was significantly improved. Based on this, Tyagi et al. [30,31,32,33,34,35] and Kumar et al. [36] analyzed and optimized the regenerative [30,32,36], intercooled regenerative [31,33] and complex [34,35] BCYs with IHPs. With dimensionless power output (DPO), TEF, dimensionless power density (DPD) [37] and dimensionless ecological function (DEF) [38] as the objectives, Wang et al. [37,38] analyzed an endoreversible BCY modified by isothermal heating and found that the power density and ecological function are the trade-offs between power output and TEF. Based on this, Tang et al. [39,40] further applied the NSGA-II algorithm to optimize an endoreversible cycle and carried out sensitivity analysis. Arora et al. [41,42] analyzed and optimized a regenerative BCY with an IHP. Based on [5], Qi et al. [43] analyzed a binary BCY with two IHPs, derived the expression of DPO and TEF, and found that the DPO can be maximized by choosing an optimal pressure ratio. Tang et al. [44] took the DPO as the objective and the heat conductance distributions (HCDs) as design variables to optimize a closed binary Brayton with IHPs.

FTT will be applied to establish a model of closed regenerative BCY with an IHP and variable-temperature heat reservoirs in this paper. The isothermal pressure drop ratio (IPDR) is variable. The DPO, TEF, DPD and DEF will be obtained, and the influences of the regenerator’s effectiveness on the performance indexes will be analyzed. Taking the DPD as an example, the optimal HCDs among heat exchangers under maximum DPD will be gained. Based on the NSGA-II algorithm, the DPO, TEF, DPD and DEF will be maximized and the corresponding Pareto frontier will be obtained. The optimal value will be chosen from the Pareto frontier by applying the LINMAP, TOPSIS and Shannon entropy methods.

2. Cycle Model

The schematic and T-s diagrams of a closed irreversible regenerative modified BCY with an IHP and coupled to variable-temperature heat reservoirs (VTHRs) are shown in Figure 1 and Figure 2, respectively. A compressor (C), a regenerator, a regular combustion chamber (RCC), a converging combustion chamber (CCC), a turbine (T) and a precooler are the main components of the cycle. The cycle consists of an irreversible adiabatic compression process (process 1–2), two isobaric heat absorption processes (processes 2–3 and 3–4), an IHP (process 4–5), an irreversible adiabatic expansion process (process 5–6) and two isobaric exothermic processes (processes 6–7 and 7–1). The processes of 1–2 s and 5–6 s are the corresponding isentropic processes.

It is assumed that the WF is a compressible ideal gas. The temperature of the WF gradually increases as it flows through a heating pipeline at subsonic speeds; the temperature decreases as it flows through a pipeline with a decreasing cross-sectional area and without heating at subsonic speeds. Combined with the two points, the convergent pipeline can be precisely and appropriately adjusted to achieve the isothermal heating of the WF [1]. Thus, the WF is isothermally heated in the CCC, and the thermal energy absorbed by the WF is converted into the kinetic energy.

The thermal capacitance rates of the WF and outer fluids at the RCC, CCC and precooler are $C_{wf}$, $C_H$, $C_{H1}$ and $C_L$, respectively. The dimensionless maximum specific capacity is $v_5/v_1$. The heat conductances of the regenerator, RCC, CCC and precooler are $U_R$, $U_H$, $U_{H1}$ and $U_L$, the numbers of heat transfer units are $N_R$, $N_H$, $N_{H1}$ and $N_L$, and the effectivenesses of the heat exchangers’ are $E_R$, $E_H$, $E_{H1}$ and $E_L$, respectively. The corresponding expressions are:

$$N_R=U_R/C_{wf},\hspace{0.17em}{N}_H=U_H/C_{Hmin},\hspace{0.17em}{N}_{H1}=U_{H1}/C_{H1},\hspace{0.17em}{N}_L=U_L/C_{Lmin}$$

(1)

$$E_R=N_R/(N_R+1)$$

(2)

$$E_H=\frac{1-\exp [-N_H(1-C_{Hmin}/C_{Hmax})]}{1-(C_{Hmin}/C_{Hmax})\exp \left[-N_H(1-C_{Hmin}/C_{Hmax})\right]}$$

(3)

$$E_{H1}=1-\exp (-N_{H1})$$

(4)

$$E_L=\frac{1-\exp [-N_L(1-C_{Lmin}/C_{Lmax})]}{1-(C_{Lmin}/C_{Lmax})\exp [-N_L(1-C_{Lmin}/C_{Lmax})]}$$

(5)

where $C_{Hmax}=max\{C_{wf},\hspace{0.17em}{C}_H\}$, $C_{Hmin}=min\{C_{wf},\hspace{0.17em}{C}_H\}$, $C_{Lmax}=max\{C_{wf},\hspace{0.17em}{C}_L\}$ and $C_{Lmin}=$ $min\{C_{wf},\hspace{0.17em}{C}_L\}$.

When $C_{Hmax}=C_{Hmin}$ or $C_{Lmax}=C_{Lmin}$, Equations (3) and (5) are simplified as:

$$E_H=N_H/(N_H+1)$$

(6)

$$E_L=N_L/(N_L+1)$$

(7)

Processes 1–2s and 5–6s are the isentropic processes, so there are:

$$T_{2s}/T_1={\pi }^m=x$$

(8)

$$T_5/T_{6s}={\pi }^m{\pi }_t^m=xy$$

(9)

where $m=(k-1)/k$ and $k$ is the ratio of specific heats.

For the IHP of process 4–5, one has:

$$T_4=T_5$$

(10)

$${\dot{Q}}_{4-5}=\dot{m}(h_5-h_4)-\dot{m}{\displaystyle {\int }_4^{5}v}dp=-\dot{m}{R}_g{T}_4\ln {\pi }_t$$

(11)

where $\dot{m}$ is the mass flow rate, $h$ is the specific enthalpy, ${\pi }_t$($=p_5/p_4\le 1$) is the IPDR, and ${\dot{Q}}_{4-5}$ is heat transfer rate. The degree of isothermal heating can be measured by the value of ${\pi }_t$. Simultaneously, ${\pi }_t$ satisfies the following constraints:

$${\pi }^{-1}\le {\pi }_t\le 1,\hspace{0.17em}\hspace{0.17em}{\pi }_t=\exp [\frac{-c_p(k-1)(M_5^2-M_4^2)}{2R_g}]$$

(12)

The velocity of the WF must be subsonic, i.e., $M_5\le 1$. Considering that the WF has an initial velocity, $M_4=0.2$ and $M_5=1$, thus ${(M_5^2-M_4^2)}_{\max }=0.96$ and ${({\pi }_t)}_{\min }=0.5107$ can be obtained.

The irreversibilities in the compressor and turbine are considered. The efficiencies of them are:

$${\eta }_c=(T_{2s}-T_1)/(T_2-T_1)$$

(13)

$${\eta }_t=(T_5-T_6)/(T_5-T_{6s})$$

(14)

The heat transfer rate (${\dot{Q}}_{2-3}$) at the regenerator, heat-absorbing rates (${\dot{Q}}_{3-4}$ and ${\dot{Q}}_{4-5}$) at the RCC and CCC and heat-releasing rate (${\dot{Q}}_{5-1}$) at the precooler are, respectively:

$${\dot{Q}}_{2-3}=C_{wf}(T_3-T_2)=C_{wf}(T_6-T_7)=C_{wf}{E}_R(T_6-T_2)$$

(15)

$${\dot{Q}}_{3-4}=C_{wf}(T_4-T_3)=C_H(T_{H1}-T_{H2})=C_{Hmin}{E}_H(T_{H1}-T_3)$$

(16)

$${\dot{Q}}_{4-5}=\dot{m}(V_5^2-V_4^2)/2=C_{H1}(T_{H3}-T_{H4})=C_{H1}{E}_{H1}(T_{H3}-T_4)$$

(17)

$${\dot{Q}}_{7-1}=C_{wf}(T_7-T_1)=C_L(T_{L2}-T_{L1})=C_{Lmin}{E}_L(T_7-T_{L1})$$

(18)

The power output ($W$), TEF ($\eta$), power density ($P$), entropy production rate ($s_g$) and ecological function ($E$) are, respectively, calculated as:

$$W={\dot{Q}}_{3-4}+{\dot{Q}}_{4-5}-{\dot{Q}}_{7-1}$$

(19)

$$\eta =W/({\dot{Q}}_{3-4}+{\dot{Q}}_{4-5})$$

(20)

$$P=W/v_6$$

(21)

$$s_g=C_H\ln (T_{H2}/T_{H1})+C_{H1}\ln (T_{H4}/T_{H3})+C_L\ln (T_{L2}/T_{L1})$$

(22)

$$E=W-T_0{s}_g$$

(23)

where $T_0$ is the ambient temperature.

In combination with Equations (8)–(10) and Equations (13)–(18), the analytical expressions of the state points’ temperatures are:

$$T_3=a_1=\frac{\begin{array}{l}{C}_{Lmin}{C}_{wf}{E}_L{T}_{L1}xy(E_R-1)({\eta }_c+x-1)-C_{Hmin}{E}_H{T}_{H1}[xy({\eta }_t-1)-\\ {\eta }_t]\{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+\\ x-1)\}\end{array}}{\begin{array}{l}{C}_{Hmin}{C}_{Lmin}{E}_H{E}_L(2E_R-1)({\eta }_c+x-1)[xy({\eta }_t-1)-{\eta }_t]-C_{Hmin}{C}_{wf}\\ E_H[E_R({\eta }_c+2x-2)-{\eta }_c-x+1][xy({\eta }_t-1)-{\eta }_t]+C_{Lmin}{C}_{wf}{E}_L\\ ({\eta }_c+x-1)[xy(E_R+{\eta }_t-2E_R{\eta }_t-1)+{\eta }_t(2E_R-1)]+C_{wf}^2\{{\eta }_t(x\\ y-1)[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]+xy(E_R+x-1-x{E}_R)\}\end{array}}$$

(24)

$$T_1=a_2=\frac{a_1{\eta }_c(E_R-1)(C_{wf}-C_{Lmin}{E}_L)-C_{Lmin}{E}_L{E}_R{\eta }_c{T}_{L1}}{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+x-1)}$$

(25)

$$T_{L2}=a_3=\frac{\begin{array}{l}{C}_{Lmin}{E}_L\{C_{wf}{\eta }_c(E_R-1)(a_1-T_{L1})-C_L{T}_{L1}(2E_R-1)({\eta }_c+x-1)+C_{wf}{T}_{L1}\\ [x-1-2E_R(x-1)]\}+C_L{C}_{wf}{T}_{L1}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]\end{array}}{C_L\{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+x-1)\}}$$

(26)

$$T_2=\frac{(x+{\eta }_c-1)[a_1{\eta }_c(C_{wf}-C_{Lmin}{E}_L)(E_R-1)-C_{Lmin}{E}_L{E}_R{\eta }_c{T}_{L1}]}{{\eta }_c\{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)(x+{\eta }_c-1)\}}$$

(27)

$$T_{2s}=\frac{x[a_1{\eta }_c(E_R-1)(C_{wf}-C_{Lmin}{E}_L)-C_{Lmin}{E}_L{E}_R{\eta }_c{T}_{L1}]}{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+x-1)}$$

(28)

$$T_4=T_5=\frac{C_{Hmin}{E}_H{a}_1+C_{Hmin}{E}_H{T}_{H1}+C_{wf}{a}_1}{C_{wf}}$$

(29)

$$T_6=\frac{(E_R-1)({\eta }_c+x-1)[a_1{\eta }_c(E_R-1)(C_{wf}-C_{Lmin}{E}_L)-C_{Lmin}{E}_L{E}_R{\eta }_c{T}_{L1}]}{E_R{\eta }_c\{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+x-1)\}}+\frac{a_1}{E_R}$$

(30)

$$T_{6s}=\frac{a_1(C_{wf}-C_{Hmin}{E}_H)+C_{Hmin}{E}_H{T}_{H1}}{C_{wf}xy}$$

(31)

$$T_7=\frac{(2E_R-1)({\eta }_c+x-1)[a_1{\eta }_c(E_R-1)(C_{wf}-C_{Lmin}{E}_L)-C_{Lmin}{E}_L{E}_R{\eta }_c{T}_{L1}]}{{\eta }_c{E}_R\{C_{wf}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-C_{Lmin}{E}_L(2E_R-1)({\eta }_c+x-1)\}}+\frac{a_1(1-E_R)}{E_R}$$

(32)

$$T_{H2}=\frac{C_H{T}_{H1}+C_{Hmin}{E}_H{a}_1-C_{Hmin}{E}_H{T}_{H1}}{C_H}$$

(33)

$$T_{H4}=\frac{E_{H1}[C_{Hmin}{E}_H(T_{H1}-a_1)+C_{wf}(a_1-T_{H3})]}{C_{wf}}+T_{H3}$$

(34)

By substituting Equations (24)–(34) into Equations (19)–(23), the DPO ($\overline{W}$), DPD ($\overline{P}$) and DEF ($\overline{E}$) can be obtained, as follows:

$$\overline{W}=\frac{W}{C_{wf}{T}_0}=\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R{\eta }_c(a_1-T_{H1})+C_{wf}{E}_R[C_{Hmin}{E}_H(T_{H1}-a_1)+C_{H1}{E}_{H1}\\ (T_{H3}-a_1)]{\eta }_c+C_{wf}^2\{a_1{\eta }_c(E_R-1)+a_2[{\eta }_c+x-1-E_R({\eta }_c+2x-2)]\}\end{array}}{C_{wf}^2{E}_R{\eta }_c{T}_0}$$

(35)

$$\eta =\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R{\eta }_c(T_{H1}-a_1)+C_{wf}{E}_R{\eta }_c[C_{Hmin}{E}_H(T_{H1}-a_1)+C_{H1}{E}_{H1}\\ (a_1-T_{H3})]+C_{wf}^2\{a_2[1-{\eta }_c-x+E_R({\eta }_c+2x-2)]-a_1{\eta }_c(E_R-1)\}\end{array}}{E_R{\eta }_c[C_{Hmin}{E}_H(C_{wf}-C_{H1}{E}_{H1})(a_1-T_{H1})+C_{H1}{C}_{wf}{E}_{H1}(a_1-T_{H3})]}$$

(36)

$$\overline{P}=\frac{\overline{W}}{v_6}=\frac{\begin{array}{l}{a}_2\{C_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R{\eta }_c(T_{H1}-a_1)+C_{wf}{E}_R{\eta }_c[C_{Hmin}{E}_H(a_1-T_{H1})+C_{H1}\\ E_{H1}(a_1-T_{H3})]+C_{wf}^2\{a_2[1-{\eta }_c-x+E_R({\eta }_c+2x-2)]-a_1{\eta }_c(E_R-1)\left\}\right\}\end{array}}{C_{wf}^2{T}_0[a_2(E_R-1)(1-{\eta }_c-x)-a_1{\eta }_c]}$$

(37)

$$\begin{array}{l}\overline{E}=\frac{E}{C_{wf}{T}_0}=\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R{\eta }_c(a_1-T_{H1})+C_{wf}{E}_R{\eta }_c[C_{Hmin}{E}_H(T_{H1}-a_1)\\ +C_{H1}{E}_{H1}(T_{H3}-a_1)]+C_{wf}^2\{a_1{\eta }_c(E_R-1)+a_2[{\eta }_c+x-E_R({\eta }_c\\ +2x-2)-1]\}\end{array}}{E_R{T}_0{\eta }_c{C}_{wf}^2}\\ -\frac{\begin{array}{l}{C}_H\ln [1+C_{Hmin}{E}_H(a_1-T_{H1})/(C_H{T}_{H1})]+C_{H1}\ln \{1+E_{H1}[a_1-T_{H3}+C_{Hmin}\\ E_H(T_{H1}-a_1)/C_{wf}]/T_{H3}\}+C_L\ln (a_3/T_{L1})\end{array}}{C_{wf}}\end{array}$$

(38)

The major differences between this paper and the [40] are the following aspects: Firstly, the irreversibilities in the compressor and turbine are considered in this paper; this is also the difference between the endoreversible model and the irreversible one. Secondly, the regenerator is added in this paper, that is, this paper studies the performance of the regenerated cycle, while [40] studied the performance of a simple cycle. Thirdly, the ecological function is studied in this paper. Fourthly, the double, triple or quadruple objective optimization is considered in this paper, but only triple objective optimization is considered in the [40]. Therefore, this paper is significantly different from the [40]. Equations (35)–(38) have a certain universality. When $C_H$, $C_{H1}$, $C_L$, $E_{H1}$, ${\eta }_c$ and ${\eta }_t$ are different values, the cycle could be simplified to different special cycles.

When ${\eta }_c={\eta }_t=1$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed endoreversible regenerative modified BCY with an IHP and coupled to VTHRs:

$$\overline{W}=\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R(a_4-T_{H1})+C_{wf}{E}_R[C_{Hmin}{E}_H(T_{H1}-a_4)+\\ C_{H1}{E}_{H1}(T_{H3}-a_4)]+C_{wf}^2\{a_4(E_R-1)+a_5[x-E_R(2x-1)]\}\end{array}}{C_{wf}^2{E}_R{T}_0}$$

(39)

$$\eta =\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R(T_{H1}-a_4)+C_{wf}{E}_R[C_{Hmin}{E}_H(T_{H1}-a_4)+\\ C_{H1}{E}_{H1}(a_4-T_{H3})]+C_{wf}^2[a_5{E}_R(2x-1)-a_4(E_R-1)-x]\end{array}}{E_R[C_{Hmin}{E}_H(C_{wf}-C_{H1}{E}_{H1})(a_4-T_{H1})+C_{H1}{C}_{wf}{E}_{H1}(a_4-T_{H3})]}$$

(40)

$$\overline{P}=\frac{\begin{array}{l}{a}_5\{C_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R(T_{H1}-a_4)+C_{wf}{E}_R[C_{Hmin}{E}_H(a_4-T_{H1})+\\ C_{H1}{E}_{H1}(a_4-T_{H3})]+C_{wf}^2\{a_5[E_R(2x-1)-x]-a_4(E_R-1)\}\end{array}}{-C_{wf}^2{T}_0[a_{5}x(E_R-1)+a_4]}$$

(41)

$$\begin{array}{l}\overline{E}=\frac{\begin{array}{l}{C}_{H1}{C}_{Hmin}{E}_H{E}_{H1}{E}_R(a_4-T_{H1})+C_{wf}{E}_R[C_{Hmin}{E}_H(T_{H1}-a_4)+C_{H1}\\ E_{H1}(T_{H3}-a_4)]+C_{wf}^2\{a_4(E_R-1)+a_5[x-E_R(2x-1)]\}\end{array}}{E_R{T}_0{C}_{wf}^2}\\ -\frac{\begin{array}{l}{C}_H\ln [1+C_{Hmin}{E}_H(a_4-T_{H1})/(C_H{T}_{H1})]+C_{H1}\ln \{1+E_{H1}[a_4-T_{H3}+\\ C_{Hmin}{E}_H(T_{H1}-a_4)/C_{wf}]/T_{H3}\}+C_L\ln (a_6/T_{L1})\end{array}}{C_{wf}}\end{array}$$

(42)

where

$$a_4=\frac{C_{Lmin}{C}_{wf}{E}_L{T}_{L1}xyx(E_R-1)+C_{Hmin}{E}_H{T}_{H1}\{C_{wf}[E_R(2x-1)-x]-C_{Lmin}{E}_{L}x(2E_R-1)\}}{\begin{array}{l}{C}_{Hmin}{C}_{wf}{E}_H[E_R(2x-1)-x]-C_{Hmin}{C}_{Lmin}{E}_H{E}_{L}x(2E_R-1)+C_{Lmin}{C}_{wf}{E}_{L}x[xy(E_R-2E_R)\\ +(2E_R-1)]+C_{wf}^2\{(xy-1)[E_R(2x-1)-x]+xy(E_R+x-1-x{E}_R)\}\end{array}}$$

(43)

$$a_5=\frac{a_1(E_R-1)(C_{wf}-C_{Lmin}{E}_L)-C_{Lmin}{E}_L{E}_R{T}_{L1}}{C_{wf}[E_R(2x-1)-x]-C_{Lmin}{E}_{L}x(2E_R-1)}$$

(44)

$$a_6=\frac{\begin{array}{l}{C}_{Lmin}{E}_L\{C_{wf}(E_R-1)(a_1-T_{L1})-C_L{T}_{L1}x(2E_R-1)+C_{wf}\\ T_{L1}[x-1-2E_R(x-1)]\}+C_L{C}_{wf}{T}_{L1}[E_R(2x-1)-x]\end{array}}{C_L\{C_{wf}[E_R(2x-1)-x]-C_{Lmin}{E}_{L}x(2E_R-1)\}}$$

(45)

When $C_H=C_L\to \infty$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed irreversible regenerative modified BCY with an IHP and coupled to constant-temperature heat reservoirs (CTHRs):

$$\overline{W}=\frac{\begin{array}{l}{C}_{H1}{E}_H{E}_{H1}{E}_R{\eta }_c(a_7-T_{H1})+E_R{\eta }_c[C_{wf}{E}_H(T_{H1}-a_7)+C_{H1}{E}_{H1}(T_{H3}-\\ a_7)]+C_{wf}\{a_7{\eta }_c(E_R-1)+a_8[{\eta }_c+x-1-E_R({\eta }_c+2x-2)]\}\end{array}}{C_{wf}{E}_R{\eta }_c{T}_0}$$

(46)

$$\eta =\frac{\begin{array}{l}{C}_{H1}{E}_H{E}_{H1}{E}_R{\eta }_c(T_{H1}-a_7)+C_{wf}\{a_8[1-{\eta }_c-x+E_R({\eta }_c+2x-2)]-\\ a_7{\eta }_c(E_R-1)\}+E_R{\eta }_c[C_{wf}{E}_H(T_{H1}-a_7)+C_{H1}{E}_{H1}(a_7-T_{H3})]\end{array}}{E_R{\eta }_c[E_H(C_{wf}-C_{H1}{E}_{H1})(a_7-T_{H1})+C_{H1}{E}_{H1}(a_7-T_{H3})]}$$

(47)

$$\overline{P}=\frac{\begin{array}{l}{a}_8\{C_{H1}{E}_H{E}_{H1}{E}_R{\eta }_c(T_{H1}-a_7)+E_R{\eta }_c[C_{wf}{E}_H(a_7-T_{H1})+C_{H1}{E}_{H1}(a_7\\ -T_{H3})]+C_{wf}\{a_8[1-{\eta }_c-x+E_R({\eta }_c+2x-2)]-a_7{\eta }_c(E_R-1)\}\}\end{array}}{C_{wf}{T}_0[a_8(E_R-1)(1-{\eta }_c-x)-a_7{\eta }_c]}$$

(48)

$$\begin{array}{l}\overline{E}=\frac{\begin{array}{l}{C}_{H1}{C}_{wf}{E}_H{E}_{H1}{E}_R{\eta }_c(a_7-T_{H1})+C_{wf}{E}_R{\eta }_c[C_{wf}{E}_H(T_{H1}-a_7)+C_{H1}{E}_{H1}\\ (T_{H3}-a_7)]+C_{wf}^2\{a_7{\eta }_c(E_R-1)+a_8[{\eta }_c+x-E_R({\eta }_c+2x-2)-1]\}\end{array}}{E_R{T}_0{\eta }_c{C}_{wf}^2}\\ -\frac{\begin{array}{l}{C}_H\ln [1+C_{wf}{E}_H(a_7-T_{H1})/(C_H{T}_{H1})]+C_{H1}\ln \{1+E_{H1}[a_7-T_{H3}+E_H\\ (T_{H1}-a_7)]/T_{H3}\}+C_L\ln (a_9/T_{L1})\end{array}}{C_{wf}}\end{array}$$

(49)

where

$$a_7=\frac{\begin{array}{l}{E}_L{T}_{L1}xy(E_R-1)({\eta }_c+x-1)-E_H{T}_{H1}[({\eta }_t-1)xy-{\eta }_t]\{[E_R({\eta }_c+2x-\\ 2)-{\eta }_c-x+1]-E_L(2E_R-1)({\eta }_c+x-1)\}\end{array}}{\begin{array}{l}{E}_H{E}_L(2E_R-1)({\eta }_c+x-1)[({\eta }_t-1)xy-{\eta }_t]-E_H[E_R({\eta }_c+2x-2)-{\eta }_c-x+\\ 1][xy({\eta }_t-1)-{\eta }_t]+E_L({\eta }_c+x-1)[xy(E_R+{\eta }_t-2E_R{\eta }_t-1)+{\eta }_t(2E_R-1)]+\\ \{{\eta }_t(xy-1)[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]+xy(-x{E}_R+E_R+x-1)\}\end{array}}$$

(50)

$$a_8=\frac{a_7{\eta }_c(E_R-1)(1-E_L)-E_L{E}_R{\eta }_c{T}_{L1}}{[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-E_L(2E_R-1)({\eta }_c+x-1)}$$

(51)

$$a_9=\frac{\begin{array}{l}{E}_L\{C_{wf}{\eta }_c(E_R-1)(a_7-T_{L1})-C_L{T}_{L1}(2E_R-1)({\eta }_c+x-1)+C_{wf}{T}_{L1}\\ [x-1-2E_R(x-1)]\}+C_L{T}_{L1}[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]\end{array}}{C_L\{[E_R({\eta }_c+2x-2)-{\eta }_c-x+1]-E_L(2E_R-1)({\eta }_c+x-1)\}}$$

(52)

When $C_H=C_L\to \infty$ and ${\eta }_c={\eta }_t=1$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed endoreversible regenerative modified BCY with an IHP and coupled to CTHRs:

$$\overline{W}=\frac{\begin{array}{l}{C}_{H1}{E}_H{E}_{H1}{E}_R(a_{10}-T_{H1})+E_R[C_{wf}{E}_H(T_{H1}-a_{10})+C_{H1}{E}_{H1}\\ (T_{H3}-a_{10})]+C_{wf}\{a_{10}(E_R-1)+a_{11}[x-E_R(2x-1)]\}\end{array}}{C_{wf}{E}_R{T}_0}$$

(53)

$$\eta =\frac{\begin{array}{l}{C}_{wf}\{a_{11}[E_R(2x-1)-x]-a_{10}(E_R-1)\}+E_R[C_{wf}{E}_H(T_{H1}-\\ a_{10})+C_{H1}{E}_{H1}(a_{10}-T_{H3})]+C_{H1}{E}_H{E}_{H1}{E}_R(T_{H1}-a_{10})\end{array}}{E_R[E_H(C_{wf}-C_{H1}{E}_{H1})(a_{10}-T_{H1})+C_{H1}{E}_{H1}(a_{10}-T_{H3})]}$$

(54)

$$\overline{P}=\frac{\begin{array}{l}{a}_{11}\{C_{H1}{E}_H{E}_{H1}{E}_R(T_{H1}-a_{10})+E_R[C_{wf}{E}_H(a_{10}-T_{H1})+C_{H1}\\ E_{H1}(a_{10}-T_{H3})]+C_{wf}\{a_{11}[E_R(2x-1)-x]-a_{10}(E_R-1)\left\}\right\}\end{array}}{-C_{wf}{T}_0[x{a}_{11}(E_R-1)+a_{10}{\eta }_c]}$$

(55)

$$\begin{array}{l}\overline{E}=\frac{\begin{array}{l}{C}_{H1}{C}_{wf}{E}_H{E}_{H1}{E}_R(a_{10}-T_{H1})+C_{wf}{E}_R[C_{wf}{E}_H(T_{H1}-a_{10})+C_{H1}\\ E_{H1}(T_{H3}-a_{10})]+C_{wf}^2\{a_{10}(E_R-1)+a_{11}[x-E_R(2x-1)]\}\end{array}}{E_R{T}_0{C}_{wf}^2}\\ -\frac{\begin{array}{l}\{C_H\ln [1+C_{wf}{E}_H(a_{10}-T_{H1})/(C_H{T}_{H1})]+C_{H1}\ln \{1+E_{H1}[a_{10}-\\ T_{H3}+C_{wf}{E}_H(T_{H1}-a_{10})/C_{wf}]/T_{H3}\}+C_L\ln (a_{12}/T_{L1})\end{array}}{C_{wf}}\end{array}$$

(56)

where

$$a_{10}=\frac{E_L{T}_{L1}{x}^{2}y(E_R-1)+E_H{T}_{H1}\{[E_R(2x-1)-x]-E_{L}x(2E_R-1)\}}{\begin{array}{l}{E}_H[E_R(2x-1)-x]+\{(xy-1)[E_R(2x-1)-x]+xy(1-x)\\ (E_R-1)\}+E_{L}x[(2E_R-1)-E_{R}xy]-E_H{E}_{L}x(2E_R-1)\end{array}}$$

(57)

$$a_{11}=\frac{a_{10}(E_R-1)(1-E_L)-E_L{E}_R{T}_{L1}}{E_R(2x-1)-x-E_{L}x(2E_R-1)}$$

(58)

$$a_{12}=\frac{\begin{array}{l}{E}_L\{C_{wf}(E_R-1)(a_{10}-T_{L1})-C_L{T}_{L1}x(2E_R-1)+C_{wf}\\ T_{L1}(x-1)(1-2E_R)\}+C_L{T}_{L1}[E_R(2x-1)-x]\end{array}}{C_L\{[E_R(2x-1)-x]-E_{L}x(2E_R-1)\}}$$

(59)

When $E_{H1}=0$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed irreversible regenerative BCY coupled to VTHRs. When $E_{H1}=0$ and ${\eta }_c={\eta }_t=1$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed endoreversible regenerative BCY coupled to VTHRs. When $E_{H1}=0$ and $C_H=C_L\to \infty$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed irreversible regenerative BCY coupled to CTHRs. When $E_{H1}=0$, $C_H=C_L\to \infty$ and ${\eta }_c={\eta }_t=1$, Equations (35)–(38) are simplified into $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ of a closed endoreversible regenerative BCY coupled to CTHRs.

3. Single Objective Analysis and Optimization

In numerical calculations, $C_{H1}=0.6\hspace{0.17em}\mathrm{kW}/\mathrm{K}$, $C_L=C_H=1.2\hspace{0.17em}\mathrm{kW}/\mathrm{K}$, $T_0=300\hspace{0.17em}\mathrm{K}$, $k=1.4$, $R_g=0.287\hspace{0.17em}\mathrm{kJ}/(\mathrm{kg}\cdot \mathrm{K})$, $E_H=E_{H1}=E_L=0.9$, $C_{wf}=1\hspace{0.17em}\mathrm{kW}/\mathrm{K}$, ${\tau }_{H1}=4.33$, ${\tau }_{H3}=5$, ${\tau }_L=1$ and ${\eta }_c={\eta }_t=0.9$ are set. The numerical solution of y is obtained by combining Equations (11), (17) and (29). The numerical solution of y is substituted into Equations (35)–(38) to obtain the numerical solutions of $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$.

3.1. Single Objective Analysis

The influences of the effectiveness ($E_R$) of regenerator on $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$, ${\pi }_t$ and $v_5/v_1$ are analyzed under the given $\pi$. The relationships of $\overline{W}$ and $\eta$, $\overline{P}$ and $\overline{E}$ versus $\pi$ with different $E_R$ are shown in Figure 3 and Figure 4, respectively. From the two figures, $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ all rise firstly and then fall as $\pi$ rises, and there are respective optimal values of $\pi$ to make $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ reach their maximum values, respectively. When $\pi <12.0$, $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ all increase as $E_R$ increases; when $\pi >12.0$, $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ all decrease as $E_R$ increases. The maximums of $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ gradually move to the left as $E_R$ increases, and the influence on $\eta$ among them is the greatest. When $\pi =12.0$ (i.e., $T_2=T_6$), increasing $\pi$ will reduce $T_3$, which allows the exhaust to carry away more heat.

Figure 5 shows the relationships of ${\pi }_t$ and $v_5/v_1$ versus $\pi$ with different $E_R$. From Figure 5, $E_R$ has a great influence on ${\pi }_t$. Given different $E_R$, the variation rule between ${\pi }_t$ and $\pi$ are different. $E_R$ has a great influence on $T_3$: when $\pi <12.0$, ${\pi }_t$ increases as $E_R$ increases, which reduces the degree of the isothermal heating; when $\pi >12.0$, ${\pi }_t$ decreases as $E_R$ increases, which aggrandizes the degree of the isothermal heating. $v_5/v_1$ decreases as $\pi$ increases. When $\pi <12.0$, $v_5/v_1$ increases as $E_R$ increases; when $\pi >12.0$, $v_5/v_1$ decreases as $E_R$ increases.

Figure 6 shows the relationships of $\overline{W}$ and $\eta$ versus $\pi$ with different ${\eta }_c$ and ${\eta }_t$. Figure 7 shows the relationships of $\overline{P}$ and $\overline{E}$ versus $\pi$ with different ${\eta }_c$ and ${\eta }_t$. It can be seen from the figure that $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ increase as ${\eta }_c$ or ${\eta }_t$ increases. As $\pi$ increase, ${\eta }_c$ or ${\eta }_t$ has a greater influence on $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$. When ${\eta }_c={\eta }_t=1$, the cycle is endoreversible, that is, the irreversibilities in the compressor and turbine is not considered. Figure 8 shows the relationships of ${\pi }_t$ and $v_5/v_1$ versus $\pi$ with different ${\eta }_c$ and ${\eta }_t$. It can be seen from the figure that ${\pi }_t$ and $v_5/v_1$ decrease as ${\eta }_c$ or ${\eta }_t$ increases. It indicates that the degree of the cycle’s isothermal heating correction is improved and the volume of the cycle device is reduced by reducing the irreversibility of the cycle.

3.2. Single Objective Optimization

3.2.1. Optimal Distributions of Heat Exchanger Inventory

In the actual design process, the heat exchanger inventory is limited or given, which indicates that the total heat conductance ($U_T$) is limited or given. According to Equations (35)–(38), $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ are related to $U_H$, $U_{H1}$, $U_L$ and $U_R$. When $U_T$ is given, the reasonable distribution of the heat conductances is conducive to improving the cycle performance. With fixed total heat conductance $U_T$($=U_H+U_{H1}+U_L+U_R$), the HCDs among the RCC, CCC, precooler and regenerator are:

$$u_H=U_H/U_T,\hspace{0.17em}{u}_{H1}=U_{H1}/U_T,\hspace{0.17em}{u}_L=U_L/U_T,\hspace{0.17em}{u}_R=U_R/U_T$$

(60)

The HCDs satisfy the following relationships:

$$u_H+u_{H1}+u_L+u_R=1$$

(61)

$$0\le u_H,\hspace{0.17em}{u}_{H1},\hspace{0.17em}{u}_L,\hspace{0.17em}{u}_R<1$$

(62)

With a given $U_T$, the HCDs are optimized to maximize $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$. Figure 9 presents the optimization flowchart for single objective. The optimization calculations of different objectives are similar. Therefore, only the optimization results of $\overline{P}$ is taken as an example for discussions. The optimization results of $\overline{W}$, $\eta$ and $\overline{E}$ are not presented herein.

Figure 10, Figure 11 and Figure 12 present the relationships of the maximum power density (${\overline{P}}_{\mathrm{opt}}$) and the corresponding power (${\overline{W}}_{{\overline{P}}_{\mathrm{opt}}}$), TEF (${\eta }_{{\overline{P}}_{\mathrm{opt}}}$), ecological function (${\overline{E}}_{{\overline{P}}_{\mathrm{opt}}}$), and other optimal parameters ${({\pi }_t)}_{{\overline{P}}_{\mathrm{opt}}}$, ${(u_H)}_{{\overline{P}}_{\mathrm{opt}}}$, ${(u_{H1})}_{{\overline{P}}_{\mathrm{opt}}}$, ${(u_L)}_{{\overline{P}}_{\mathrm{opt}}}$ and ${(u_R)}_{{\overline{P}}_{\mathrm{opt}}}$ versus $\pi$. From Figure 10, ${\overline{W}}_{{\overline{P}}_{\mathrm{opt}}}$ and ${\overline{P}}_{\mathrm{opt}}$ increase first and then decrease as $\pi$ increases. When $\pi \le 9.2$, ${\eta }_{{\overline{P}}_{\mathrm{opt}}}$ first augments and then slightly decreases as $\pi$ augments; When $\pi >9.2$, ${\eta }_{{\overline{P}}_{\mathrm{opt}}}$ first augments and then decreases as $\pi$ augments.

From Figure 11, when $\pi \le 9.2$, ${\overline{E}}_{{\overline{P}}_{\mathrm{opt}}}$ augments first and then slightly augments as $\pi$ augments; when $\pi >9.2$, ${\overline{E}}_{{\overline{P}}_{\mathrm{opt}}}$ augments first and then slightly decreases as $\pi$ augments. ${({\pi }_t)}_{{\overline{P}}_{\mathrm{opt}}}$ first decreases and then augments as $\pi$ augments, and the degree of the isothermal heating first augments and then decreases. ${(u_H)}_{{\overline{P}}_{\mathrm{opt}}}$ decreases as $\pi$ augments. From Figure 12, when $\pi \le 9.2$ and $\pi$ increases, ${(u_{H1})}_{{\overline{P}}_{\mathrm{opt}}}$ diminishes first and then increases slightly, ${(u_L)}_{{\overline{P}}_{\mathrm{opt}}}$ increases rapidly, and ${(u_R)}_{{\overline{P}}_{\mathrm{opt}}}$ drops rapidly; when $\pi >9.2$ and $\pi$ increases, ${(u_{H1})}_{{\overline{P}}_{\mathrm{opt}}}$ decreases first and then increases, ${(u_L)}_{{\overline{P}}_{\mathrm{opt}}}$ slowly increases, and ${(u_R)}_{{\overline{P}}_{\mathrm{opt}}}$ is always 0.

In combination with the previous performance analysis, the regenerator is meaningless when $\pi$ exceeds a certain value. In this case, $E_R$ should be 0, and the cycle is simplified to a closed irreversible modified simple BCY with an IHP and coupled to VTHRs. In short, the effect of the regenerator is better when $\pi$ is relatively small; it is not recommended to add a regenerator to the cycle when $\pi$ is relatively large.

3.2.2. Optimal Thermal Capacitance Rate Matching among the WF and Heat Reservoir

The influences of the thermal capacitance rate matching among the WF and heat reservoir on the cycle performance are discussed. Figure 13 illustrates the relationships of ${\overline{P}}_{\mathrm{opt}}$ versus $C_H/C_{wf}$ and $C_{H1}/C_{wf}$ when $\pi =8$. The figure shows that ${\overline{P}}_{\mathrm{opt}}$ first increases and then tends to remain constant as $C_H/C_{wf}$ increases. Because when $C_H<C_{wf}$, $C_{H\max }=C_{wf}$ and $C_{H\min }=C_H$; when $C_H>C_{wf}$, $C_{H\max }=C_H$ and $C_{H\min }=C_{wf}$; ${\overline{P}}_{\mathrm{opt}}$ first increases and then tends to remain constant as $C_{H1}/C_{wf}$ increases. Because ${\dot{Q}}_{3-4}$ first increases and then tends to remain constant as $C_{H1}/C_{wf}$ increases. To further increase the DPD, it is necessary to increase ${\dot{Q}}_{2-3}$, that is, to increase $u_H$ or $U_H$. But as ${\dot{Q}}_{2-3}$ increases, ${\dot{Q}}_{3-4}$ will also increase, which forms a balance between the two. Finally, the DPD tends to remain constant. By numerical calculations, ${\overline{W}}_{{\overline{P}}_{\mathrm{opt}}}$, ${\eta }_{{\overline{P}}_{\mathrm{opt}}}$ and ${\overline{E}}_{{\overline{P}}_{\mathrm{opt}}}$ also first increase and then tend to remain constants as $C_H/C_{wf}$ or $C_{H1}/C_{wf}$ increases. Thus, there is an optimal $C_H/C_{wf}$ or $C_{H1}/C_{wf}$ to make ${\overline{W}}_{{\overline{P}}_{\mathrm{opt}}}$, ${\eta }_{{\overline{P}}_{\mathrm{opt}}}$, ${\overline{P}}_{\mathrm{opt}}$ and ${\overline{E}}_{{\overline{P}}_{\mathrm{opt}}}$ to reach the optimal values.

Figure 14 illustrates the relationship of ${\overline{P}}_{\mathrm{opt}}$ versus $C_H/C_{wf}$ with different $C_L/C_H$. Figure 15 illustrates the relationship of ${\overline{P}}_{\mathrm{opt}}$ versus $C_{H1}/C_{wf}$ with different $C_L/C_{H1}$. As can be seen from the figures, when $C_H/C_{wf}$ or $C_{H1}/C_{wf}$ is smaller, ${\overline{P}}_{\mathrm{opt}}$ increases as $C_L/C_H$ or $C_L/C_{H1}$ increases; when $C_H/C_{wf}$ or $C_{H1}/C_{wf}$ is larger, ${\overline{P}}_{\mathrm{opt}}$ is unaffected by $C_L/C_H$ or $C_L/C_{H1}$. By adjusting the value of $C_L/C_H$ or $C_L/C_{H1}$, the value of $C_H/C_{wf}$ or $C_{H1}/C_{wf}$ could be changed when ${\overline{P}}_{\mathrm{opt}}$ reaches the optimal values. Under the condition that the DPD is kept constant, the requirement of the heat exchangers corresponding to the high-temperature reservoirs can be reduced by increasing $C_L/C_H$ or $C_L/C_{H1}$ and reducing $C_H/C_{wf}$ or $C_{H1}/C_{wf}$.

4. Multi-Objective Optimization

Figure 16 shows the optimization flowchart for calculating the Pareto frontier using the NSGA-II algorithm [39,40,41,45,46,47,48,49,50,51,52,53,54]. With $\pi$ and HCDs as the design variables and $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ as the optimization objectives, the cycle’s double, triple or quadruple objective optimization is conducted.

The Pareto frontier contains many feasible solutions, which need to be selected by some methods. The LINMAP, TOPSIS and Shannon entropy methods are three common decision-making methods [40]. These three methods are used to get the optimal scheme from the Pareto frontier, and the results are compared. Finally, the merits and demerits of each decision method are judged by the deviation index (D). Positive ideal point (PIP) is the optimal value combination of four objective functions in the Pareto frontier while negative ideal point (NIP) is the worst value combination. The LINMAP method, TOPSIS method and D are all based on the PIP and NIP.

For the Shannon entropy method, it is assumed that the Pareto frontier contains n feasible solutions, and each feasible solution contains m objectives. $F_{ij}$ ($1\le i\le m$, $1\le j\le n$) represents the value of the j-th optimization objective for the i-th feasible solution, and $i_{opt}$ is the best possible solution in Equation (63).

$$i_{opt}\in \min \{P_{ij}\cdot w_j\}$$

(63)

$$P_{ij}=F_{ij}/{\displaystyle \sum _{i=1}^n{F}_{ij}},\hspace{0.17em}S{E}_j=-\frac{1}{\ln n}{\displaystyle \sum _{i=1}^n{P}_{ij}}\ln P_{ij}$$

(64)

$$w_j=(1-S{E}_j)/{\displaystyle \sum _{j=1}^n(1-S{E}_j)}$$

(65)

where $P_{ij}$ represents the dimensionless value of $F_{ij}$, and $S{E}_j$ and $w_j$ are the Shannon entropy and the weight of j-th optimization objective, respectively.

For LINMAP method, $i_{opt}$ is the best possible solution in Equation (66).

$$i_{\mathrm{opt}}\in \min \left\{E{D}_{i+}\right\}$$

(66)

$$E{D}_{i+}=\sqrt{{\displaystyle {\sum }_{j=1}^m{(F_{ij}-F_j^{positive})}^2}}$$

(67)

where $E{D}_{i+}$ represents the Euclidean distance between the i-th feasible solution and the PIP, and $F^{positive}$ represents the value of the PIP.

For the TOPSIS method, $i_{opt}$ is the best possible solution in Equation (68).

$$i_{\mathrm{opt}}\in \max \left\{\frac{E{D}_{i+}}{E{D}_{i+}+E{D}_{i-}}\right\}$$

(68)

$$E{D}_{i-}=\sqrt{{\displaystyle {\sum }_{j=1}^m{(F_{ij}-F_j^{negative})}^2}}$$

(69)

where $E{D}_{i-}$ represents the distance between the i-th feasible solution and the NIP, and $F^{negative}$ represents the value of the NIP.

The deviation index is calculated according to Equation (70):

$$D=\frac{\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_j-F_j^{positive})}^2}}}{\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_j-F_j^{positive})}^2}}+\sqrt{{\displaystyle {\sum }_{j=1}^m{(G_j-F_j^{negative})}^2}}}$$

(70)

where $G_j$ represents the value of the j-th optimization objective corresponding to the decision method.

Figure 17 shows the Pareto frontier with $\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$ as the optimization objectives. The line’s color of the Pareto frontier represents the value of the DEF.

Table 1. Comparison of double, triple and quadruple objective optimization’s results.

Optimization Methods	Decision Methods	Optimization Variables					Performance Indexes				Deviation Index	IPDR
		${\mathit{u}}_{\mathit{H}}$	${\mathit{u}}_{\mathit{H}1}$	${\mathit{u}}_{\mathit{R}}$	${\mathit{u}}_{\mathit{L}}$	$\mathit{\pi }$	${\overline{\mathit{W}}}_{\mathbf{max}}$	${\mathit{\eta }}_{\mathbf{max}}$	${\overline{\mathit{P}}}_{\mathbf{max}}$	${\overline{\mathit{E}}}_{\mathbf{max}}$	D	${\mathit{\pi }}_{\mathit{t}}$
Quadruple objective ($\overline{W}$, $\eta$, $\overline{P}$ and $\overline{E}$) optimization	LINMAP	0.251	0.180	0.002	0.567	14.009	0.786	0.395	0.381	0.458	0.259	0.567
	TOPSIS	0.251	0.180	0.002	0.567	14.009	0.786	0.395	0.381	0.458	0.259	0.567
	Shannon Entropy	0.236	0.142	0.106	0.517	7.480	0.799	0.390	0.344	0.467	0.283	0.563
Triple objective ($\overline{W}$, $\eta$ and $\overline{P}$) optimization	LINMAP	0.234	0.172	0.003	0.591	14.596	0.782	0.397	0.381	0.455	0.268	0.559
	TOPSIS	0.234	0.172	0.003	0.591	14.596	0.782	0.397	0.381	0.455	0.268	0.559
	Shannon Entropy	0.221	0.142	0.115	0.522	7.316	0.798	0.392	0.341	0.466	0.285	0.552
Triple objective ($\overline{W}$, $\eta$ and $\overline{E}$) optimization	LINMAP	0.197	0.162	0.0009	0.641	13.909	0.786	0.397	0.378	0.457	0.254	0.530
	TOPSIS	0.197	0.162	0.0009	0.641	13.909	0.786	0.397	0.378	0.457	0.254	0.530
	Shannon Entropy	0.224	0.139	0.103	0.534	7.644	0.799	0.390	0.345	0.466	0.281	0.554
Triple objective ($\overline{W}$, $\overline{P}$ and $\overline{E}$) optimization	LINMAP	0.248	0.166	0.0002	0.586	13.643	0.792	0.395	0.380	0.463	0.237	0.568
	TOPSIS	0.248	0.166	0.0002	0.586	13.643	0.792	0.395	0.380	0.463	0.237	0.568
	Shannon Entropy	0.241	0.153	0.0003	0.606	12.751	0.798	0.391	0.378	0.466	0.238	0.562
Triple objective ($\eta$, $\overline{P}$ and $\overline{E}$) optimization	LINMAP	0.262	0.155	0.004	0.580	14.934	0.777	0.397	0.380	0.452	0.291	0.585
	TOPSIS	0.262	0.155	0.004	0.580	14.934	0.777	0.397	0.380	0.452	0.291	0.585
	Shannon Entropy	0.238	0.140	0.103	0.519	7.529	0.799	0.389	0.345	0.467	0.283	0.565
Double objective optimization ($\overline{W}$ and $\eta$)	LINMAP	0.208	0.1564	0.0005	0.635	15.225	0.777	0.402	0.380	0.453	0.273	0.547
	TOPSIS	0.209	0.1558	0.0005	0.634	15.119	0.778	0.402	0.379	0.454	0.269	0.548
	Shannon Entropy	0.220	0.147	0.0007	0.631	14.160	0.785	0.398	0.379	0.459	0.250	0.556
Double objective optimization ($\overline{W}$ and $\overline{P}$)	LINMAP	0.245	0.174	0.0007	0.580	12.662	0.798	0.390	0.379	0.465	0.242	0.559
	TOPSIS	0.245	0.174	0.0007	0.580	12.662	0.798	0.390	0.379	0.465	0.242	0.559
	Shannon Entropy	0.247	0.171	0.0006	0.582	11.757	0.804	0.385	0.376	0.466	0.252	0.557
Double objective optimization ($\overline{W}$ and $\overline{E}$)	LINMAP	0.206	0.1486	0.0007	0.646	16.561	0.763	0.406	0.378	0.445	0.335	0.555
	TOPSIS	0.206	0.1486	0.0007	0.646	16.561	0.763	0.406	0.378	0.445	0.335	0.555
	Shannon Entropy	0.206	0.1486	0.0007	0.646	16.561	0.763	0.406	0.378	0.445	0.335	0.555
Double objective optimization ($\eta$ and $\overline{P}$)	LINMAP	0.207	0.159	0.0007	0.633	21.169	0.715	0.413	0.371	0.406	0.630	0.570
	TOPSIS	0.207	0.159	0.0004	0.634	21.248	0.715	0.414	0.371	0.407	0.624	0.570
	Shannon Entropy	0.208	0.160	0.0005	0.634	21.249	0.715	0.415	0.372	0.408	0.618	0.570
Double objective optimization ($\eta$ and $\overline{E}$)	LINMAP	0.219	0.162	0.0003	0.619	14.589	0.784	0.400	0.380	0.458	0.247	0.551
	TOPSIS	0.221	0.163	0.0003	0.618	14.293	0.786	0.398	0.380	0.460	0.245	0.551
	Shannon Entropy	0.224	0.160	0.0003	0.616	12.766	0.797	0.391	0.378	0.465	0.240	0.548
Double objective optimization ($\overline{P}$ and $\overline{E}$)	LINMAP	0.198	0.174	0.0002	0.627	22.134	0.706	0.415	0.369	0.399	0.674	0.563
	TOPSIS	0.198	0.174	0.0002	0.627	22.134	0.706	0.415	0.369	0.399	0.674	0.563
	Shannon Entropy	0.198	0.174	0.0002	0.627	22.134	0.706	0.415	0.369	0.399	0.674	0.563
PIP		——	——	——	——	——	0.809	0.438	0.381	0.467	——	——
NIP		——	——	——	——	——	0.677	0.369	0.360	0.373	——	——

lists the comparison of double, triple and quadruple objective optimization results. For quadruple objective optimization, the results gained by LINMAP and TOPSIS methods are the same, and D is smaller than that gained by the Shannon entropy method. It indicates that the results gained by the LINMAP method are better than those gained by the Shannon entropy method. The $u_R$ gained by LINMAP or TOPSIS method is extremely small and tends to 0. The IPDRs of the three decision methods have little difference, thus the isothermal heating degrees of the three methods have little difference. $\overline{W}$ and $\overline{P}$ gained by LINMAP or TOPSIS method are larger than those gained by the Shannon entropy method, but $\eta$ and $\overline{E}$ gained by LINMAP or TOPSIS method are smaller than those gained by Shannon entropy method. For triple objective optimization, the results gained by LINMAP and TOPSIS methods are the same. The triple objective ($\overline{W}$, $\overline{P}$ and $\overline{E}$) optimization’ deviation index gained by LINMAP or TOPSIS method is the smallest. For double objective optimization, all of $u_R$ corresponding to the double objective optimization are extremely small and tend to 0.

5. Conclusions

FTT is applied to establish a model of a closed irreversible regenerative BCY with an IHP and variable IPDR in this paper. The DPO, TEF, DPD and DEF are obtained, and the influences of $E_R$ on the performance indexes are analyzed. With $\overline{P}$ as the optimization objective and HCDs as the design variables, the single objective analysis and optimization of the cycle are conducted. Based on the NSGA-II algorithm, the DPO, TEF, DPD and DEF are maximized with $\pi$ and HCDs as the design variables and the corresponding Pareto frontier is obtained. Three decision methods are applied to get the best scheme. The main conclusions manifest that:

(1)

For the single objective analysis, the DPO, TEF, DPD and DEF all augment first and then reduce as $\pi$ augments. When $\pi <12.0$, it is beneficial to add the regenerator to improve the cycle performance; when $\pi >12.0$, adding the regenerator will reduce the cycle performance. For the single-objective optimization, when $\pi >9.2$, the effectiveness at the regenerator should be 0, and the cycle will be simplified into a closed irreversible modified BCY with an IHP and coupled to VTHRs. Hence, the regenerator is added, and a suitable pressure ratio should be selected in the limited range. In the actual process, the pressure ratio should be strictly controlled in this new-type gas turbine, or the new-type gas turbine is suitable for the situations where the pressure ratio is relatively low. The DPO, TEF, DPD and DEF increase as ${\eta }_c$ or ${\eta }_t$ increases. ${\pi }_t$ and $v_5/v_1$ decrease as ${\eta }_c$ or ${\eta }_t$ increases. It indicates that the cycle performance and the degree of the cycle’s isothermal heating correction are improved and the volume of the cycle device is reduced by reducing the irreversibility of the cycle.

(2)

For the quadruple objective optimization, the results gained by LINMAP and TOPSIS methods are the same, and D is smaller than that gained by the Shannon entropy method. Therefore, the results gained by LINMAP and TOPSIS methods are better than those gained by the Shannon entropy method. For the double, triple or quadruple objective optimization, the triple objective ($\overline{W}$, $\overline{P}$ and $\overline{E}$) optimization’ deviation index gained by LINMAP or TOPSIS method is the smallest. According to decision-makers’ actual needs, they could choose the best scheme from the Pareto frontier, design the new-type gas turbine reasonably, and then determine the construction or the operation of the gas turbine.

FTT is a powerful theoretical tool and the NSGA-II algorithm is an effective mathematical tool for the performance analysis and optimization of closed irreversible regenerative BCY with an IHP and variable IPDR. The results obtained in this paper have some guiding significance for the design of a new-type gas turbine.