Exergetic sustainability evaluation and optimization of an irreversible Brayton cycle performance

Abstract

Owing to the energy demands and global warming issue, employing more effective power cycles has become a responsibility. This paper presents a thermodynamical study of an irreversible Brayton cycle with the aim of optimizing the performance of the Brayton cycle. Moreover, four different schemes in the process of multi-objective optimization were suggested, and the outcomes of each scheme are assessed separately. The power output, the concepts of entropy generation, the energy, the exergy output, and the exergy efficiencies for the irreversible Brayton cycle are considered in the analysis. In the first scheme, in order to maximize the exergy output, the ecological function and the ecological coefficient of performance, a multi-objective optimization algorithm (MOEA) is used. In the second scheme, three objective functions including the exergetic performance criteria, the ecological coefficient of performance, and the ecological function are maximized at the same time by employing MOEA. In the third scenario, in order to maximize the exergy output, the exergetic performance criteria and the ecological coefficient of performance, a MOEA is performed. In the last scheme, three objective functions containing the exergetic performance criteria, the ecological coefficient of performance, and the exergy-based ecological function are maximized at the same time by employing multi-objective optimization algorithms. All the strategies are implemented via multi-objective evolutionary algorithms based on the NSGAII method. Finally, to govern the final outcome in each scheme, three well-known decision makers were employed.

Keywords

entropy generation / exergy / Brayton cycle / ecological function / irreversibility

Introduction

Carnot efficiency was defined as the maximum theoretical efficiency of a heat engine operated between thermal reservoirs at temperatures TH and TL. Carnot machine was a totally reversible machine in opposition with real machines which were characterized by the intern and extern irreversibilities [¹]. For this reason, Carnot efficiency was not sufficient to assess the performance of the real machine [²]. With the idea of creating a model including irreversibilities, the finite time thermodynamics approach (FTT) occurred [³] in order to provide more realistic and applicable results for engineering design guidance [⁴–⁶]. After developing the FTT, some performance evaluation criteria were suggested [⁷,⁸]. Angulo-Brown [⁹] developed a function named ecological function (ECF)ECF=\dot{W}-T_L{\dot{S}}_{gen}. He compared minimum entropy production criteria and ECF and established that ECF appreciably reduced entropy production and improved power production by 10% for the Curzon Alborn cycle. Yan [¹⁰] pointed out that the loss power was T_{\mathrm{0}}{\dot{S}}_{gen}and he arranged the ECF as

ECF=\dot{W}-T_{\mathrm{0}}{\dot{S}}_{gen}

. Many irreversible cycles were investigated by using the ECF. Ust [¹¹–¹²], Ust and Sahin [¹³], and Ust et al. [¹⁴–²³] proposed a novel ecological objective function called the ecological coefficient of performance (ECOP). The ecological coefficient was defined as the power output per unit loss rate of availability ECOP={\displaystyle \frac{\dot{W}}{T_{\mathrm{0}}{\dot{S}}_{gen}}}. Ust et al. [¹⁵] analyzed an irreversible dual cycle cogeneration system based on a new exergetic performance criterion. They defined the exergetic performance coefficient (EPC) as the ratio of the total exergy flow to the loss rate of availability (EPC={\displaystyle \frac{\dot{E}x}{T_{\mathrm{0}}{\dot{S}}_{gen}}}) [¹⁶–²³]. Açıkkalp [²⁴], Açıkkalp and Yamık [²⁵] examined the maximum/minimum available work output/input of actual thermal cycles. They suggested a new criterion named maximum available work (MAW) [²⁴,²⁵]. Based on these criteria they determined the optimum points. These points were based on the maximum work or exergy and minimum work lost.

One of the eminent tools in solving engineering problems is multi-objective optimization [²⁶–³⁰]. Özyer et al. [²⁶] suggested a method for calculating the prospect calculation via evolutionary algorithms. Blecic et al. [²⁸] proposed a decision support system named Bay MODE based on the Bayesian analysis and multi-objective optimization.

Solving a multi-objective issue for the best implementation is a hard mission since it requires concurrent fulfillment of different and occasionally conflicting objective functions [²⁹,³⁰]. The multi-objective optimization method was extensively employed in energy system engineering exponentially [³¹–⁵⁹].

In this paper, four optimization scenarios were defined for the irreversible Brayton cycle. In the first scenario, in order to maximize the exergy flow, the ecological function, and the ECOP, a MOEA was executed. In the second scheme, three objective functions comprising the exergetic performance, the ECOP, and the ecological function were maximized. In the third scenario, a MOEA was executed in order to maximize the exergy flow, the exergetic performance, and the ECOP. In the last scenario, three objective functions, i.e., the exergetic performance, the ECOP, and the exergy-based ecological function were maximized.

Thermodynamic analysis

The T-s diagram of the irreversible Brayton cycle investigated is depicted in Fig. 1.

Air, as working fluid of thermodynamic cycles, can be assumed as an ideal gas with a constant specific heat for small temperature differences. However, for high-temperature differences, this assumption causes incorrect results. According to Ref. [⁶⁰], Eq. (1) introduces the specific heat at constant pressure dependence on temperature for the range of 300 K–3500 K.

(1)\begin{array}{l}{c}_p=\mathrm{2}.\mathrm{5}\mathrm{0}\mathrm{6}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{1}\mathrm{1}}{T}^{\mathrm{2}}+\mathrm{1}.\mathrm{4}\mathrm{5}\mathrm{4}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}{T}^{\mathrm{1}.\mathrm{5}}-\mathrm{4}.\mathrm{2}\mathrm{4}\mathrm{6}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}T+\mathrm{3}.\mathrm{1}\mathrm{6}\mathrm{2}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}{T}^{\mathrm{0}.\mathrm{5}}\\ +\mathrm{1}.\mathrm{3}\mathrm{3}\mathrm{0}\mathrm{3}-\mathrm{1}.\mathrm{5}\mathrm{1}\mathrm{2}\times \mathrm{1}{\mathrm{0}}^{\mathrm{4}}{T}^{-\mathrm{1}.\mathrm{5}}+\mathrm{3}.\mathrm{0}\mathrm{6}\mathrm{3}\times \mathrm{1}{\mathrm{0}}^{\mathrm{5}}{T}^{-\mathrm{2}}-\mathrm{2}.\mathrm{2}\mathrm{1}\mathrm{2}\times \mathrm{1}{\mathrm{0}}^{\mathrm{7}}{T}^{-\mathrm{3}}.\end{array}

For the ideal gas assumption, Eq. (2) is valid.

(2)c_p=c_v+R.

Using Eq. (2), c_v may be described as

(3)\begin{array}{l}{c}_v=\mathrm{2}.\mathrm{5}\mathrm{0}\mathrm{6}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{1}\mathrm{1}}{T}^{\mathrm{2}}+\mathrm{1}.\mathrm{4}\mathrm{5}\mathrm{4}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}{T}^{\mathrm{1}.\mathrm{5}}-\mathrm{4}.\mathrm{2}\mathrm{4}\mathrm{6}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}T+\mathrm{3}.\mathrm{1}\mathrm{6}\mathrm{2}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}{T}^{\mathrm{0}.\mathrm{5}}\\ +1\mathrm{.0433}-1\mathrm{.512}\times 10^4{T}^{-1\mathrm{.5}}+3\mathrm{.063}\times 10^{\mathrm{5}}{T}^{-\mathrm{2}}-\mathrm{2}.\mathrm{2}\mathrm{1}\mathrm{2}\times \mathrm{1}{\mathrm{0}}^{\mathrm{7}}{T}^{-\mathrm{3}}.\end{array}

To facilitate the calculations involving the above-defined expressions c_v and c_p, approximations with an accuracy of 0.9911 for c_v and an accuracy of 0.9914 for c_p in the range of 300 K–2500 K are expressed as

(4)c_v=0\mathrm{.2517}{T}^{0\mathrm{.1764}},

(5)c_p=0\mathrm{.4582}{T}^{0\mathrm{.1319}}.

A regression analysis for c_v and c_p is depicted in Figs. 2 and 3, correspondingly.

Equations (4) and (5) can be obtained from Eqs. (1) and (3), respectively, by means of curve fitting to the obtained values.

The heat flow received by the system can be calculated as

(6){\dot{Q}}_{\mathrm{H}}=\dot{m}{\displaystyle {\int }_{\mathrm{2}}^{\mathrm{3}}{c}_{\mathrm{p}}\mathrm{d}T},

while the heat flow rejected by the system is determined by its absolute value as

(7){\dot{Q}}_{\mathrm{L}}=\dot{m}{\displaystyle {\int }_{\mathrm{1}}^{\mathrm{4}}{c}_{\mathrm{p}}\mathrm{d}T}.

Thus, the mechanical power provided by the system can be calculated by the difference

(8)\dot{W}={\dot{Q}}_H-{\dot{Q}}_L.

The energy efficiency of the system is the ratio between the mechanical power and the heat flow to the hot source.

(9)\eta ={\displaystyle \frac{\dot{W}}{{\dot{Q}}_H}}.

The entropy generation flow due to the internal and external irreversibilities is expressed as

(10){\dot{S}}_{gen}={\displaystyle \frac{{\dot{Q}}_L}{T_{\mathrm{1}}}}-{\displaystyle \frac{{\dot{Q}}_H}{T_{\mathrm{3}}}}.

In terms of exergy, the fuel of the system is considered the exergy flow received by the cycle on its hot source [⁷]. Thus, exergy input (kW) can be expressed as

(11)\dot{E}{x}_{Q_H}={\dot{Q}}_H(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{0}}}{T_{\mathrm{3}}}}),

where T₀ is the environment temperature.

The functional diagram of the Brayton cycle (Fig. 4) shows different flows exchanged by the working gas and its surrounding in terms of energy, entropy, and exergy.

Thus, the performance of the Brayton cycle may be estimated using a more accurate indicator [⁷], the exergy efficiency

(12)n_{ex}={\displaystyle \frac{\dot{W}}{\dot{E}{x}_{Q_H}}}.

The internal irreversibilities in the Brayton cycle defined via expansion and compression efficiencies defined as

(13)\begin{array}{l}{\eta }_C={\displaystyle \frac{T_{\mathrm{2}S}-T_{\mathrm{1}}}{T_{\mathrm{2}}-T_{\mathrm{1}}}},\\ {\eta }_E={\displaystyle \frac{T_{\mathrm{4}}-T_{\mathrm{3}}}{T_{\mathrm{4}S}-T_{\mathrm{3}}}}.\end{array}

Employing Eq. (13), T₂ and T₄ are specified as

(14)T_{\mathrm{2}}={\displaystyle \frac{(T_{\mathrm{2}S}-T_{\mathrm{1}})+{\eta }_C{T}_{\mathrm{1}}}{{\eta }_C}},

(15)T_{\mathrm{4}}={\eta }_E(T_{\mathrm{4}S}-T_{\mathrm{3}})+T_{\mathrm{3}}.

In this paper, in addition to exergy flow, four ecological parameters are considered in the optimization problem.

Ecological function (ECF) [⁶¹–⁶³]

(16)ECF=\dot{W}-T_{\mathrm{0}}{\dot{S}}_{gen}.

Exergy-based ecological function (EECF)

(17)EECF=\dot{E}x-T_{\mathrm{0}}{\dot{S}}_{gen}.

Ecological coefficient of performance (ECOP)

(18)ECOP={\displaystyle \frac{\dot{W}}{T_{\mathrm{0}}{\dot{S}}_{gen}}}.

Exergetic performance criteria (EPC)

(19)EPC={\displaystyle \frac{\dot{E}x}{T_{\mathrm{0}}{\dot{S}}_{gen}}}.

The maximum available work per unit time (kW) is calculated as

(20)MAW=\dot{E}x-T_{\mathrm{0}}{\dot{S}}_{gen}.

Multi-objective optimization (MOEA)

The MOEA was employed to optimize the Brayton engine system to specify the system design variables. The flow-chart of the multi-objective method is depicted in Fig. 5 [³¹–⁴⁹].

Objective functions, limitations, and decision parameters

The exergy flow, ecological function, and ECOP are the three objective functions for the first strategy formulated through Eqs. (12), (16), and (18).

The ECOP, and the exergetic performance criteria are the three objective functions for the second strategy, which are evaluated via Eqs. (16), (18), and (19).

The exergy flow, ECOP, and exergetic performance criteria are three objective functions for the third strategy formulated through via Eqs. (12), (18), and (19).

The exergy-based ecological function, ECOP, and exergetic performance criteria are the three objective functions for the last strategy formulated through Eqs. (17), (18), and (19).

Following limitations were considered:

(21)\mathrm{0.85}\le {\eta }_C\le \mathrm{1},

(22)\mathrm{0.85}\le {\eta }_E\le \mathrm{1},

(23)\mathrm{5}\le x\le \mathrm{12},

(24)\mathrm{300}\le T_{\mathrm{1}}\le \mathrm{350}K,

(25)\mathrm{900}\le T_{\mathrm{3}}\le \mathrm{1200}K.

TOPSIS, LINMAP, and Fuzzy decision makers were used to specify the final result of each strategy. Description of these methods can be found in the previous works [²⁷–⁴⁵].

Results and discussion

As illustrated in Fig. 6(a), the exergy input of the system decreases considerably with increasing the pressure ratio (x) at different T₁. At a given pressure ratio (x), by increasing T₁, the amount of exergy flow decreases.

As depicted in Fig. 6(b), the dimensionless ecological function increases considerably with increasing the pressure ratio (x) at different T₁. When x<8 at a given pressure ratio, by increasing T₁, the amount of ECF increases. However, for a pressure ratio greater than 8, by increasing T₁, the amount of ECF decreases.

As shown in Fig. 6(c), the exergy-based ecological function (EECF) decreases with the increasing of the pressure ratio (x) at different T₁. For a pressure ratio greater than 6.5, by increasing T₁, the amount of EECF decreases.

As exhibited in Fig. 6(d), the ECOP increases significantly with increasing the pressure ratio (x) at different T₁. At a given pressure ratio (x), by increasing T₁, the amount of ECOP increases.

As illustrated in Fig. 6(e), the exergetic performance criterion increases considerably with the increasing of the pressure ratio (x) at different T₁. At a given pressure ratio (x), by increasing T₁, a number of EPC increases.

The sensitivity of the objective functions to the decision variables was examined. As depicted in Fig. 7(a), the exergy output of the system increases significantly with increasing T₃ at different T₁. As illustrated in Fig. 7(b), the ecological function increases considerably with increasing the T₃ at different T₁. As shown in Fig. 7(c), the exergy-based ecological function (EECF) increases with the increasing of the T₃ at different T₁. As depicted in Fig. 7(d), the ECOP decreases significantly with increasing the T₃ at different T₁.

As illustrated in Fig. 7(d), the exergetic performance criterion decreases considerably with increasing the T₃ at different T₁.

Results of the first scenario

Using MOEA coupled with the NSGA-II approach, the exergy flow, the ecological function and the ECOP were maximized simultaneously. The objective functions in the optimization, and the restraints employed in this scenario are expressed via Eqs. (12), (16) and (18), and Eqs. (21)–(25) respectively. The compression efficiency, the expansion efficiency, the pressure ratio, the T₁, and T₃ were assumed as design parameters throughout the optimization process. Following properties of the irreversible Brayton cycle system are presumed as [⁶⁴]

R=\mathrm{0.287}\mathrm{kJ}\cdot {\mathrm{kg}}^{-\mathrm{1}},T_{\mathrm{0}}=\mathrm{298.15}\mathrm{K},\dot{m}=\mathrm{1}\mathrm{kg}\cdot s^{-\mathrm{1}}.

The Pareto optimal frontier for three objective functions, the objective function associated with the exergy output, the ecological function, and the ECOP of the irreversible Brayton cycle are represented in Fig. 8.

Table 1 lists the optimum results achieved for objective functions and decision parameters via the TOPSIS, Fuzzy and LINMAP approaches for the first scenario.

Table 2 tabulates the error evaluation of the used decision makers. Two different statistical error indexes including MAAE (maximum absolute percentage error) and MAPE (mean absolute percentage error) were employed to calculate the errors of each decision makers. Table 2 presents the MAAE and MAPE of each decision makers.

Results of the second scenario

Three strategic objective functions were considered for optimization including the ecological function, the ECOP, and the exergetic performance criteria (should be maximized) which were formulated via Eqs. (16), (18), and (19) correspondingly.

The objective functions in this scenario were formulated via Eqs. (16), (18), and (19) and limitations are expressed by citEqs. (21)–(25).

The properties of the irreversible Brayton cycle were assumed as [⁶⁴]

R=\mathrm{0.287}\mathrm{kJ}\cdot {\mathrm{kg}}^{-\mathrm{1}},T_{\mathrm{0}}=\mathrm{298.15}\mathrm{K},\dot{m}=\mathrm{1}\mathrm{kg}\cdot s^{-\mathrm{1}}.

Figure 9 shows the Pareto frontier in the space of the suggested objectives achieved in the optimization strategy. Three final solutions were selected by the dicision-makers indicated in Fig. 9.

Table 3 gives the optimum results gained from the TOPSIS, Fuzzy and LINMAP approaches for the second scenario.

Table 4 demonstrates the error evaluation of the used decision makers. Two different statistical error indexes including MAAE and MAPE were employed to calculate the errors of each decision makers. Table 4 presents the MAAE and MAPE of each decision makers.

Results of the third scenario

Using MOEA coupled with the NSGA-II approach, the exergy output, the ECOP, and the exergetic performance criteria were maximized simultaneously. The objective functions in the optimization, and the restraints employed in this scenario are expressed via Eqs. (12), (18) and (19), and Eqs. (21)–(25) respectively. The compression efficiency, the pressure ratio, the expansion efficiency, T₁ and T₃ were considered as design parameters throughout the optimization process. The properties of the irreversible Brayton cycle system are presumed as [⁶⁴]

R=\mathrm{0.287}\mathrm{kJ}\cdot {\mathrm{kg}}^{-\mathrm{1}},T_{\mathrm{0}}=\mathrm{298.15}\mathrm{K},\dot{m}=\mathrm{1}\mathrm{kg}\cdot s^{-\mathrm{1}}.

The Pareto optimal frontier for the three objective functions, the objective function associated with the exergy output, the ECOP, and the exergetic performance criteria of the irreversible Brayton cycle are represented in Fig. 10. Table 5 lists the optimum results gained from the decision makers for the third scenario. Table 6 demonstrates the error evaluation of the used decision makers. Two different statistical error indexes including MAAE and MAPE were employed to calculate the errors of each decision makers. Table 6 presents the MAAE and MAPE of each decision makers.

Results of the last scenario

Using MOEA coupled with the NSGA-II approach, the exergy-based ecological function and ECOP and exergetic performance criteria were maximized simultaneously. The objective functions in the optimization, and the restraints employed in this scenario are expressed via Eqs. (17), (18) and (19), and citEqs. (21)-(25) respectively. The compression efficiency, the pressure ratio, the expansion efficiency, T₁ and T₃ were presumed as design variables in the optimization process. The features of the irreversible Brayton cycle system are presumed as [⁶⁴]

R=\mathrm{0.287}\mathrm{kJ}\cdot {\mathrm{kg}}^{-\mathrm{1}},T_{\mathrm{0}}=\mathrm{298.15}\mathrm{K},\dot{m}=\mathrm{1}\mathrm{kg}\cdot s^{-\mathrm{1}}.

The Pareto optimal frontier for the three objective functions, the objective function associated with the exergy-based ecological function, the ECOP, and the exergetic performance criteria of the irreversible Brayton cycle are represented in Fig. 11. Table 7 gives the optimum results achieved for objective functions and decision parameters via the TOPSIS, Fuzzy and LINMAP approaches for the last scenario. Table 8 demonstrates the error evaluation of the used decision makers. Two different statistical error indexes including MAAE and MAPE were employed to calculate the errors of each decision makers. Table 8 presents the MAAE and MAPE of each decision makers.

Conclusions

In this work, a thermodynamic analysis was performed on an irreversible Brayton cycle. The effects of the pressure ratio, the compression efficiency, the expansion efficiency, the source and sink temperatures were included in the examination of the exergy flow, the ecological function, the exergetic performance criteria, the exergy-based ecological function, and the ecological coefficient of performance of an irreversible Brayton cycle by employing thermodynamic analysis. Moreover, optimal values of the objective functions comprising the exergy flow, the ecological function, the exergetic performance criteria, the exergy-based ecological function, and the ecological coefficient of the performance were indicated.

In the MOEA approach, five separate parameters including the compression efficiency, the pressure ratio, the expansion efficiency, T₁ and T₃ were assumed as decision variables. To specify a final answer from the outcomes achieved from MOEA, three well-known decision makers were used and the final results were compared based on the error analysis.

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