Power Optimization of a Modified Closed Binary Brayton Cycle with Two Isothermal Heating Processes and Coupled to Variable-Temperature Reservoirs

: A modified closed binary Brayton cycle model with variable isothermal pressure drop ratios is established by using finite time thermodynamics in this paper. A topping cycle, a bottoming cycle, two isothermal heating processes and variable-temperature reservoirs are included in the new model. The topping cycle is composed of a compressor, a regular combustion chamber, a converging combustion chamber, a turbine and a precooler. The bottoming cycle is composed of a compressor, an ordinary regenerator, an isothermal regenerator, a turbine and a precooler. The heat conductance distributions among the six heat exchangers are optimized with dimensionless power output as optimization objective. The results show that the double maximum dimensionless power output increases first and then tends to be unchanged while the inlet temperature ratios of the regular combustion chamber and the converging combustion chamber increase. There also exist optimal thermal capacitance rate matchings among the working fluid and heat reservoirs, leading to the optimal maximum dimensionless power output.


Introduction
Due to the characteristics of high power density (PD), small vibration, high automation, low operating pressure and easy lubrication, gas turbine plants (Brayton heat engine cycle) are extensively applied in the fields of aviation, energy, transportation, etc. According to the different working fluid (WF) circulation modes, the Brayton cycle is divided into open and closed cycles, and many works concerning the classical thermodynamic analyses and optimizations for various Brayton cycles have been performed [1][2][3]. The WF of closed Brayton cycle is not directly connected with the atmosphere, and does not participate in the combustion process. Hence, it is applied to convert nuclear energy, geothermal energy, solid fuel and other primary energy into electricity energy.
For the case of simple heating, the temperature of the gas elevates in the direction of a duct when the subsonic compressible gas flows through the smooth heating duct with a fixed cross-sectional area. For the case of simple cross-sectional area change, the temperature drops when the gas flows through the smooth adiabatic duct with a reduced cross-sectional area. Based on these two gas properties, the isothermal processes can be realized when the subsonic compressible gas flows through the smooth heating duct with a reduced cross-sectional area. Vecchiarelli et al. [4] presented a combustion chamber where the WF could be heated isothermally. The introduction of this type of combustion chamber could effectively improve the thermal efficiency (TEF) of the Brayton cycle, and reduce the emissions of nitrogen oxides and other harmful gases. Göktun and Yavuz [5] applied the isothermal heating combustion chamber to the regenerative Brayton cycle, and discovered the isothermal pressure drop ratio impacted the cycle performance significantly. Based on [5], Erbay et al. [6] compared the optimal performances of an isothermal heating regenerative Brayton cycle under maximum power output (PO) and maximum PD. Jubeh [7] found the exergy efficiency was enhanced by adding the isothermal heating combustion chamber in regenerative Brayton cycle. El-Maksoud [8] combined the isothermal concept and double Brayton cycle to establish a new cycle model. Based on [8], Qi et al. [9] derived the specific work and exergy efficiency, and analyzed the impacts of different parameters on the exergy efficiency. All those works were carried out by using classical thermodynamics.
Based on El-Maksoud's classical thermodynamic model [8], Qi et al. [62] established a closed endoreversible binary Brayton cycle model with two isothermal processes, without internal irreversibility, and coupled to constant-temperature reservoirs (CTRs). They derived the functional expressions of PO, TEF, PD and ecological function, respectively. The impacts of different thermodynamic parameters on the relationships among performance indexes and the pressure ratio of the topping cycle were analyzed, and the heat conductance distributions (HCDs) among heat exchangers were further optimized.
Based on the previously established cycle models in [8,62], a modified closed binary Brayton cycle (MCBBC) with variable isothermal pressure drop ratios and internal and external irreversibilities will be established by using FTT in this paper. The HCDs will be optimized with dimensionless PO. The influences of different thermodynamic parameters on the optimal performance will be analyzed, and the thermal capacitance rate matchings (TCRMs) among the WF and the heat reservoirs will be also discussed. Figure 1a shows the schematic diagram of the MCBBC with two isothermal heating processes [8]. A topping cycle and a bottoming cycle are included in the model. The topping cycle is composed of a compressor (Com1), a regular combustion chamber (RCC), a converging combustion chamber (CCC), a turbine (Tur1) and a precooler (PC1). The bottoming cycle is composed of a compressor (Com2), an ordinary regenerator (OR), an isothermal regenerator (IR), a turbine (Tur2) and a precooler (PC2).

Cycle Model
When (3) and (4) are simplified into: The efficiencies in the Com1, Com2, Tur1 and Tur2 are com1 η , com2 η , tur1 η and tur2 η , respectively: The heat absorbing rates of WF in the RCC, CCC, OR and IR are 2 3 The power output and thermal efficiency of the cycle are W and η , respectively: Processes 3-4 and 3a-4a are the isothermal ones, and the heat absorbing rates of the two processes are: Processes 1-2s, 1-2as, 4-5s and 4a-5as are the isentropic processes; namely:  where x , y , a x and a y are the parameters of temperature ratios which can be calculated by the pressure ratios. The upper limits of t π and ta π are 1, which means that the isothermal heating is not used.
When 1 t π < or 1 ta π < , the topping or bottoming cycle adopts the isothermal process. The isothermal pressure drop ratios must meet the following constraints: otherwise, the cycle is meaningless. The major difference between the model in this paper and that in [8] is that all the heat transfer losses in the six heat exchangers are considered in this paper. This is also the major difference between classical thermodynamic model and FTT model. The major differences between the model in this paper and that in [62] are two aspects: One is that the irreversible compression and expansion losses are considered in this paper; this is also the major difference between the endoreversible model and the irreversible one. Another is that the heat reservoirs are assumed to be variable-temperature ones in this paper; this is one of basic characteristics of practical closed engineering cycles.
According to the above model, the dimensionless PO W and TEF η can be given by: where 1 a , 2 a , 3 a , 4 a and 5 a are listed in Appendix A.
If Equations (21) and (22) are not considered when solving Equations (28) and (29), the final analytical solutions for W and η of the cycle cannot be obtained. Considering Equations (21) and (22), the values of x and y in Equations (28) and (29) are obtained by numerical calculation, and the corresponding values of W and η can be obtained.
different values, the model can be converted into different special models; Equations (28) and (29) can be reduced to the corresponding dimensionless PO and TEF respectively, which have a certain universality. When com1 com2 tur1 tur2 1 η = η = η = η = , Equations (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed endoreversible binary Brayton cycle with two isothermal heating processes and coupled to variable-temperature heat reservoirs (VTHRs):   (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed irreversible binary Brayton cycle with two isothermal heating processes and coupled to CTRs: where 1 c , 2 c , 3 c , 4 c and 5 c are listed in Appendix A.
When com1 com2 tur1 tur2 (28) and (29) (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed irreversible Brayton cycle with an isothermal heating process and coupled to VTHRs: where e is listed in Appendix A. When (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed irreversible Brayton cycle with an isothermal heating process and coupled to CTRs: (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed endoreversible Brayton cycle with an isothermal heating process and coupled to VTHRs [59,60]: (28) and (29) can be reduced to the dimensionless PO and TEF of a modified closed endoreversible Brayton cycle with an isothermal heating process and coupled to CTRs: (28) and (29) can be reduced to the dimensionless PO and TEF of a closed irreversible simple Brayton cycle coupled to VTHRs [37,63]: (28) and (29) can be reduced to the dimensionless PO and TEF of a closed irreversible simple Brayton cycle coupled to CTRs [37,63]: (28) and (29) can be reduced to the dimensionless PO and TEF of a closed endoreversible simple Brayton cycle coupled to VTHRs [64]: (28) and (29) can be reduced to the dimensionless PO and TEF of the closed endoreversible simple Brayton cycle coupled to CTRs [65,66]:

Optimal Heat Conductance Distributions
With W as the optimization objective, the heat conductances of six heat exchangers will be optimized by fixing the total heat conductance (THC); namely, The HCDs in the RCC, CCC, OR, IR, PC1 and PC2 are defined as: where j u must meet the following constraints: The dimensionless PO of the MCBBC can be maximized by optimizing the HCDs. Finally, the maximum dimensionless PO ( max W ) and the corresponding optimal HCDs (  Table 1. The flow chart of dimensionless PO optimization is displayed in Figure 2. Calculate the negative number of the dimensionless power output, and then the function "fmincon" in MATLAB is used to solve the minimum value. The parameters of "fmincon" are: "TolCon" is 10 −6 , "TolConSQP" is 300 and "TolFun" is 10 −6 .  π ) in the Com1 and Com2, respectively.
Thermal capacity rate of outer fluid at PC1 Thermal capacity rate of outer fluid at PC2 Inlet temperature ratio of outer fluid at RCC increases. There is a set of optimal pressure ratios ( can only be used to maintain the operation of the bottoming cycle. Figure 4a shows that    According to the numerical calculation, the influences of the temperature ratios, compressor efficiencies, turbine efficiencies and THC on optimization results are further analyzed. Figure 7a,

Optimal Thermal Capacitance Rate Matchings
The W of the cycle also affected by the TCRMs. By taking W as the optimization objective, TCRMs are optimized. Figure 9a,