Introduction

Nowadays, fossil fuel consumption has increased with an increase in population and the growth of various industries. This, in turn, has increased the emission of various environmental pollutants [1]. To address this issue, researchers seek ways to replace fossil fuels and improve the efficiency of energy-consuming systems. In this regard, there are two major solutions. The first solution is the use of renewable energies [2]. The second solution involves optimizing and increasing the efficiency of existing systems, for which various methods have been presented [2]. Research and experience have shown that multi-generation systems, such as combined cooling, heating, and power systems, can significantly improve efficiency and reduce emissions [3]. In addition, recovering the waste heat in various industries and refineries in order to operate multi-generation systems can greatly increase the efficiency of these systems [4, 5]. The recovered waste heat depends on the quality of the dissipated thermal energy and the energy demand of the system and can be utilized for generating power, cooling, fresh water, etc. [6,7,8]. A major parameter in the selection of the heat recovery system type is the quality of the dissipated energy, which is influenced by the flow rate and temperature of the gas. Figure 1 displays the share of different heat generation sources used in waste heat recovery (WHR) systems [9].

Fig. 1
figure 1

Percent share of different heat sources used in a WHR system

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The Brayton cycle can be used up to a temperature of 1000 °C depending on the application and hence, creates good flexibility in selecting the heat source [10]. This cycle can be either open or closed [11]. In an open cycle, the working fluid is air, which provides the energy required by a turbine by burning with fuel [12]. In this case, other gases, such as helium, CO2, etc. cannot be used as the working fluid. Moreover, due to the discharge of combustion products to the environment, this cycle is not suitable in terms of energy and environmental efficiency. As a result, it is often used in combination with other cycles [13]. On the other hand, the closed Brayton cycle does not discharge the working fluid to the environment and, thus, allows designers to examine different working fluids. However, this cycle also needs additional systems to improve its energy efficiency due to the high temperature of the fluid at the turbine outlet.

To this end, numerous studies have been conducted, and researchers have investigated various hybrid systems. Naserian et al. [14] studied the cycle introduced by Alali and Al-Shaboul [15] and launched a closed Brayton cycle using waste heat from a nuclear reactor. They also used the exhaust gases from the Brayton cycle to produce steam and heating. Four working fluids were studied, namely N2, He, CO2, and air. According to their results, He performed best in power generation, while N2 and air produced a higher specific power. Moreover, CO2 exhibited superior performance in steam generation (heating system). In another study, Romano and Riberio [16] examined and optimized the cold-side temperature of the Brayton cycle. The cycle they studied transferred the heat from the turbine outlet flow to the pipes via a heat exchanger and further to the environment via radiators for heating purposes. The ratio of the generated power to the surface area of the radiators was considered the objective function. The results indicated that this ratio was maximum when the temperature of the heat pipes was between 450 and 500 K. Wang et al. [17] tried to combine an internal combustion engine, closed Brayton cycles, refrigeration, and organic Rankine cycle (ORC). Also, in this research, using the single objective optimization method, the compressor inlet temperature and pressure ratio, the Brayton turbine inlet temperature, the ORC turbine inlet pressure and the ejector outlet pressure have been obtained as 3, 36.242 C, 435.404 C, 1.363 MPa, and 1.556 MPa, respectively. Abbasi and Pourrahmani [18] studied a hybrid system consisting of the Brayton cycle, ORC, and an absorption chiller using multi-objective optimization and eco-exergy methods by considering solar energy as a heat source. They employed a (phase change material) PCM tank for using the system at night. As such, the system could work 24 h a day by storing solar energy by the ammonia storage source, during the day and utilizing it at night. Moreover, the reverse osmosis (RO) desalination system was operated by using the power generated by the ORC system. Based on the results, in the optimal state, the pressure ratio of the Brayton cycle compressor, the turbine inlet pressure, the exergy efficiency, and the energy efficiency were 9.06, 3300 kPa, 14.40%, and 40.5%, respectively. Furthermore, the generated amounts of power, cooling, and water were obtained to be, respectively, 2.42 MW, 1.6 MW, and 5209.5 m3/day. In another study, Yongming Feng et al. [19] conducted a thermodynamic and optimization study on a hybrid Brayton-Kalina cycle. In this research, the diesel generator exhaust was considered to be the heat source. According to the results, the annual fuel cost decreased by 16.62%, and the energy efficiency improved by 15.01%. Ahmadi et al. [20] studied the combination of an open Brayton cycle with the ORC and absorption chiller cycles. The system proposed in this research was capable of heating, cooling, and power cogeneration. The results showed that the highest irreversibility and exergy destruction occurred in the combustion chamber and the heat exchanger. Moreover, a study of the emitted CO2 indicated that emissions of this gas had been reduced due to the simultaneous use of different systems. Extensive research has been carried out in the field of WHR, the most important of which studied the combination of cycles such as Brayton, ORC, absorption chiller, and desalinations. Some of this research has already been mentioned in Table 1.

Table 1 Comparative presentation of prior integrated cogeneration systems with respect to power generation, cooling capacity and the maximum temperature
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In 2020, Liu et al. investigated the Brayton cycle for power generation systems. This study aimed to minimize the total mass of the power system through the optimization of the influencing parameters of the system components. The results showed that increasing the turbine inlet temperature by 4% from the value of 1150 K leads to a 6% reduction in the mass of the system [26]. In 2020, Kim et al. did work on a parametric study and optimization of a closed Brayton cycle considering fluid charging. The results showed that the amount of charge can be considered as a design and control parameter of CBC [27]. In 2020, Zhang et al. investigated a new power generation system using a gas turbine cycle. Based on the optimization results, the thermal and exergy efficiency of the system was reported as 46.11% and 47.24% [28]. In 2022, Musharavati et al. investigated biomass gasification for electricity and freshwater production. The main components of this power plant include a gasifier, a compressor, a heat exchanger, a gas turbine, a combustion chamber, and a multi-effect desalination unit with thermal vapor compression [29]. In 2023, Khanmohammadi et al. investigated an energy-based multigeneration system. The heliostat had the highest exergy destruction rate with 1867 kW. Pump and heliostat units had the lowest exergy efficiency with values of 52.09% and 65.39% [30]. In 2023, Ding et al. evaluated a multiple-generation multi-generation energy system. In this research, a new system consisting of the Kalina cycle, organic Rankine cycle, refrigeration cycle, and electrolyzer was evaluated. The proposed system can produce 80.1 kW and 1930 g/h of electricity and hydrogen fuel, respectively [31]. In 2022, Colakoglu and Durmayaz worked to investigate a new multiple energy generation system based on a solar tower with a triple combined power cycle. Under optimal conditions, the system had energy and exergy efficiency values of 51.99% and 37.99%, respectively, and contributed to 513.8 kg of carbon dioxide emissions reduction per hour [32].

The use of energy systems is necessary to improve consumption and increase the performance of energy systems. With the increase in the world's population, as well as the decrease in fossil resources, and on the other hand, the increase in environmental pollution, more attention should be paid to energy production systems. For this reason, in recent years, many studies have been conducted in the field of increasing the performance of energy production systems with various methods, including heat recovery from the system, optimization, integration of several systems together, simultaneous production, etc. [33,34,35,36,37,38,39,40,41,42,43,44].

Today, the energy crisis, the increase in greenhouse gas emissions and global warming are critical challenges in the world, especially in developing countries, and the optimal and correct use of energy can be a suitable solution for these problems. For this reason, a lot of research has been done in this field, but more research is needed [45,46,47,48,49,50,51,52,53,54,55].

Nowadays, it is inevitable to implement new power plants with high efficiency to deal with environmental challenges. Because the world's need for energy is increasing day by day, high-capacity systems for production must be set up. One of the most common power generation cycles around the world is the Brayton cycle, which uses natural gas. Energy consumption is one of the appropriate criteria for determining the level of progress and quality of life in a country. Continuity of energy supply and the possibility of long-term access to resources require a comprehensive energy planning and for this reason energy planning is considered an undeniable economic, national and strategic necessity. Energy demand is increasing day by day in the world. Energy has a significant impact on the ability of industries to produce, the type of tools and machines, the methods of exploitation and transportation. Man has become so dependent on energy that he rarely even bothers to think about its role and impact.

Today, meeting energy needs with the least environmental impact is a serious challenge that must be solved with knowledge and technology. Therefore, the development of economic energy systems that provide the maximum use of non-renewable energy sources for electricity production is of great importance. In this regard, in this research, a system for producing clean electricity is proposed to deal with this issue. The need for this system is recommended especially in different climates to meet the need for electrical energy.

The present study examines a multi-generation system for the simultaneous production of electricity and cooling through waste heat. The studied system consists of a closed Brayton cycle whose waste heat is used by an ORC, leading to improved energy efficiency. In addition, excess heat from the exhaust that cannot be used in the Brayton cycle enters the absorption chiller cycle to provide cooling. The purpose of this research is to increase system performance and system productivity.

In short, the current work activity is as follows:

  • Introducing a new system of combining Brayton cycle units, organic Rankine cycle with the aim of maximum power generation.

  • Using the waste heat of the Brayton cycle to supply the energy required for the Rankine organic cycle.

  • Using energy, exergy and exergeoeconomic analyzes to provide a comprehensive view of system performance and capabilities.

  • Provide a sensitivity analysis to evaluate the impact of design variables on system performance for a deeper understanding of the method.

  • Multi-objective optimization of the system to increase the performance of the proposed system.

  • Selection of three objective functions of exergy efficiency, energy efficiency and system production power.

  • Using the TOPSIS multi-criteria decision making method to find the best and most optimal Pareto point that represents the best performance of the system.

System description

The present research examines the feasibility of using waste heat from the exhaust flare of a refinery via a combination of the closed Brayton cycle, the ORC, and the absorption chiller cycle. Based on the information collected from the refineries in Iran, the flow rate of gas flare varies from 7 to 13 kg s−1. In addition, its temperature can reach 1000 °C. Accordingly, the flow rate of the hot fluid entering the heat exchanger is considered to be 10 kg s−1 for modeling the system and extracting the results. The proposed system is displayed in Fig. 2. In this system, the Brayton cycle consists of two heat exchangers (heater and reheater), high-pressure (HP) and low-pressure (LP) turbines, and a compressor. Moreover, the absorption chiller and the ORCs are of simple types. The working fluid of the Brayton cycle is variable. As can be seen, the exhaust flare first enters the Brayton cycle and loses heat to the working fluid. Then, it transfers part of its energy to the Brayton cycle again by passing through the reheater. Subsequently, cooling is generated as the gas passes through the generator of the absorption chiller cycle. Finally, the gas is discharged from the system at a minimum temperature of 120 °C. In the Brayton cycle, the fluid gains temperature after passing through the compressor and receiving heat in the heat exchanger (this increase in temperature depends on the temperature of the heat source). The fluid enters the first turbine at Point 3, where some of its thermal energy is converted to power. Then, it enters the reheater at Point 4, where its temperature increases at constant pressure. At Point 5, it enters the LP turbine, re-producing power. Given that the fluid must be cooled before entering the compressor, an ORC is used to reduce the temperature of the working fluid of the Brayton cycle and prevent heat loss. Hence, the temperature of the fluid drops as it exchanges heat with the ORC. This process continues again.

Fig. 2
figure 2

A schematic of the proposed system

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As mentioned previously, the outlet fluid of the LP turbine enters the ORC power generation cycle, where heat is exchanged, leading to a reduction in the temperature of the Brayton cycle fluid before it enters the compressor and generates power, which improves the system performance. Figure 3 displays the T-S graph of the ORC with benzene as the working fluid.

Fig. 3
figure 3

T-S graph of the ORC with benzene as the working fluid

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Methodology

This section describes the modeling of the system equipment. To this end, first, the assumptions made for the thermodynamic modeling and analysis are presented. The assumptions are as follows [56, 57]:

  • The system operates at a steady-state condition.

  • Temperature and pressure heat loss are ignored.

  • Changes in the kinetic and potential energies are ignored for exergy calculations.

  • The ambient temperature and pressure are considered to be 25 °C and 101 kPa, respectively.

  • Pressure losses in pipelines and heat exchangers can be neglected.

  • The isentropic efficiency of compressors, gas turbines, ORC turbines, and pumps are 85, 87, and 90%, respectively.

  • Ideal gas mixture law is applied and considered for synthesis gas, fuel and combustion due to high temperatures in gasifier and combustion chamber.

The exhaust gas flow rate is considered to be 10 kg s−1 in this research. Other parameters affecting the design are presented in Table 2.

Table 2 Parameters for designing the multi-generation system [58]
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Closed Brayton cycle

The cycle used in this research is the closed Brayton cycle, which makes use of waste heat. As shown in Fig. 2, this system generates power by absorbing the exhaust heat in two stages (heater and preheater) and using LP and HP turbines. Moreover, the power generated by the turbine and compressor are determined by Eq. (3) and (4), respectively [59, 60].

$$Q_{{{\text{HX}}1}} = m_{{{\text{wf}}}} \left( {h_{4} - h_{3} } \right)$$
(1)
$$Q_{{{\text{HX}}2}} = m_{{{\text{wf}}}} \left( {h_{6} - h_{5} } \right)$$
(2)
$$W_{{{\text{Tur}}_{{\text{k}}} }} = \eta_{{{\text{Tur}}}} \cdot m_{{{\text{wf}}}} \left( {\Delta h} \right)$$
(3)
$$W_{{{\text{Comp}}}} = \frac{{m_{{{\text{wf}}}} \left( {\Delta h} \right)}}{{\eta_{{{\text{Comp}}}} }}$$
(4)

In these equations, \({\eta }_{\mathrm{Comp}}\) and \({\eta }_{\mathrm{Tur}}\) represent the isentropic efficiency of the compressor and turbine, respectively (Table 2). Also, the following equation is used to compute the cross-sectional area of the heat exchangers [36, 42].

$$A_{{\text{k}}} = \frac{{Q_{{\text{k}}} }}{{U_{{\text{k}}} \cdot \Delta T_{{{\text{lm}}_{{\text{k}}} }} }}$$
(5)

where Uk is the overall heat transfer coefficient. Table 3 presents the overall heat transfer coefficient for all the heat exchangers [61, 62].

Table 3 Overall heat transfer coefficient for all the heat exchangers [35,36,37,38,39,40,41,42,43,44,45,46,47]
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ORC unit

Due to the closed nature of the gas cycle, a cooling tower is required to reduce the temperature of the outlet fluid from the second turbine via the exchange of heat. However, this increases the energy losses in the system. Hence, the waste heat from the gas turbine cycle is transferred to the ORC via a heat exchanger by eliminating the cooling tower. In addition, the power consumption of its pump is obtained from Eq. (5) [35, 41, 42].

$$W_{{{\text{pump}}}} = m_{{{\text{ORC}}}} .\frac{{\upsilon_{7} \left( {P_{1} - P_{7} } \right)}}{{\eta_{{\text{p}}} }}$$
(6)

Here, \(\eta_{{{\text{pump}}}}\) denotes the isentropic efficiency of the pump. Equation (7) is used to calculate this value [35, 41, 42].

$$\eta_{{\text{pump }}} = \frac{{75 + 11.5\log_{10} \left( {W_{{\text{p }}} } \right) - 1.5\left[ {\log_{10} \left( {W_{{\text{p}}} } \right)} \right]^{2} }}{100}$$
(7)

Absorption chiller

Given the flow rate and temperature of the exhaust after the preheater, it is possible to recover the waste heat. A single-effect absorption chiller was used for this purpose. Moreover, the working fluid is a lithium-bromide/water mixture. As shown in Fig. 2, the thermal energy enters the absorption chiller cycle via a generator. Hence, the maximum temperature occurs in the generator. The fluid entering the generator, which is a mixture of water and lithium, receives heat, part of its water evaporates, and it is divided into two parts: water vapor and a concentrated mixture. The water vapor enters the condenser and is converted to saturated liquid after the exchange of heat (with an external source at a lower temperature). This liquid enters a pressure valve and loses pressure as the temperature decreases to 5 °C at constant enthalpy. This temperature can vary depending on the application. In addition, the concentrated mixture enters the heat exchanger first due to its high temperature and preheats the backflow from the absorber. Then, it passes through the pressure-relief valve, reaches the pressure of the evaporator inlet water, and enters the absorber. In the absorber, water is reabsorbed by the lithium-bromide mixture. This absorption process is exothermic; hence, heat is exchanged with an external heat source. Equation (8) determines the heat entering the absorption chiller [34, 40, 62].

$$Q_{{{\text{in}}_{{\text{generator}}} }} = m_{{{\text{ex}}}} \left( {h_{{{\text{ex}}_{{3}} }} - h_{{{\text{ex}}_{4} }} } \right) = m_{{{\text{apn}}_{7} }} h_{{{\text{apn}}_{7} }} + m_{{{\text{apn}}_{3} }} - m_{{{\text{apn}}_{4} }} h_{{{\text{apn}}_{4} }}$$
(8)

The heat transfer surface in the generator, condenser, evaporator, absorber, and solution heat exchangers can be calculated in a similar manner to Eq. (8). The heat transfer coefficients are also shown.

Exergy analysis

Exergy refers to the part of energy that can be used in different forms. Exergy balance was used to calculate the exergy destruction in different equipment. Equation (9) represents the exergy balance [40, 42].

$${\text{Ex}}_{{{\text{in}}_{{\text{k}}} }} - {\text{Ex}}_{{{\text{out}}_{{\text{k}}} }} + {\text{Ex}}_{{{\text{destruction}}_{{\text{k}}} }} = \frac{{\delta {\text{Ex}}_{{\text{k}}} }}{\delta t}$$
(9)

In this equation, \(\frac{{\delta {\text{Ex}}}}{\delta t} = 0\) since steady-state conditions are considered. Table 4 presents the exergy balance for different equipment. Moreover, Eq. (10) is used to compute the exergy efficiency [40, 42].

$$\eta_{{{\text{ex}}}} = 1 - \left( {\frac{{{\text{Ex}}_{{{\text{destruction}}}} }}{{{\text{Ex}}_{{{\text{in}}}} }}} \right)$$
(10)
Table 4 Exergy balance for different component [33,34,35,36,37,38,39,40,41,42,43,44]
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Coefficient of performance

Coefficient of performance or COP represents the efficiency of absorption chiller cooling devices. The COP performance coefficient is obtained by the ratio of thermal power taken from the air of the desired space or water and any other fluid to the input electrical power or the sum of electrical power and input thermal power. The performance factor is the same as the efficiency of machines, but its value is more than 100%. If its value is less than 100% for a device, that device will not work properly. That's why COP is greater than 1 or 100%, which, while converting all work into heat, pumps heat from a source to where it is needed, and this COP greater than 1 does not mean that The desired device has an efficiency higher than 100%. Transferring heat from one place to another requires less energy than converting work into heat. The absorption chiller coefficient of performance (COP) is calculated from Eq. 11 [63,64,65,66,67]:

$${\text{COP}} = \frac{Q}{W}$$
(11)

Optimization

Currently, many problems in various fields require optimization and the selection of an optimal point due to the conflict between the behavior of different parameters. The optimal point is selected based on the significance of different parameters, and no single parameter is the sole basis for selection. In this regard, multi-objective optimization methods are widely used in different areas. The Pareto solution method is a suitable multi-objective optimization method with acceptable performance. There are various techniques, such as linear programming technique for multidimensional analysis of preference (LINMEP), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), etc. for selecting the optimal point in the Pareto method. The present research used the TOPSIS technique. In this method, the point furthest from the worst point and nearest to the best point is selected as the optimal point.

TOPSIS is a multi-indicator decision-making method for evaluating and prioritizing options based on criteria according to their distance from positive and negative ideals. The underlying logic of this method defines the ideal solution and the negative ideal solution. The ideal solution is the one that increases the benefit criterion and decreases the cost criterion. The optimal option is the option that has the smallest distance from the ideal solution and at the same time the farthest distance from the negative ideal solution. In the ranking of options by the TOPSIS method, the options that are most similar to the ideal solution are ranked higher. The target space between two criteria is shown as an example in the figure. Here, A + and A − are the ideal solution and the negative ideal solution, respectively [67, 68].

In this research, compressor pressure ratio and intermediation pressure ratio were considered as decision variables, three target functions of total power generation, energy efficiency and exergy were considered as target functions to improve system performance.

The Pareto diagram is drawn to show the simultaneous optimization of the objective functions and to find the best value of the objective functions. In this research, three objective functions of energy efficiency, exergy efficiency and production power were considered as optimization functions to improve system performance, and the purpose of drawing a three-dimensional optimization diagram, which is called Pareto diagram, is to find the best Pareto point, which it shows the most optimal performance of the system. A Pareto chart is used to highlight the most important factors among a set of factors. This diagram in the quality control process often shows the causes of the most common defects or the reason for the most common customer complaints. After finding a range of optimal points with the multi-objective optimization method and drawing a Pareto diagram, the TOPSIS method, which is a decision-making method for finding the best and worst points, finds the most optimal Pareto point among different points (Fig. 4).

Fig. 4
figure 4

A schematic of the system modeling and optimization

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Table 5 presents the range of decision making variables.

Table 5 Optimization variables and their range
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Results and discussion

The variable parameters of the system are inlet pressure and pressure ratios of compressor and turbines in the Brayton cycle, and the maximum pressure of the ORC. Accordingly, these parameters are first examined for helium as the working fluid and a heat source with a temperature of 800 °C. Subsequently, the effects of these parameters on the power and cooling generation are investigated, and the optimal value of each parameter is determined. In the first step, the objective is to select an optimal compressor inlet pressure. A rise in the compressor’s pressure does not significantly affect its power consumption and the working fluid’s temperature due to its pressure ratio being constant. Hence, the variations in the system due to an increase in the inlet pressure of the Brayton cycle are negligible. Figure 5 shows the effect of the compressor inlet pressure on the total generated work and cooling. As seen in the figure, increasing the compressor pressure ratio from 100 to 1000 has reduced the total power of the system from 2409.5 to 2403.5 kW, which has caused a 0.24% reduction in the system's production power rate, which is very small and insignificant. Also, the results show that the increase in the compressor pressure ratio from 100 to 1000 has caused an increase in the production rate of the system from 5739.8 to 5743.2 kW, which has caused a 0.059% increase in the production cooling rate of the system, which is considered very little. Thus, the initial pressure was considered to be 100 kPa.

Fig. 5
figure 5

The effect of the compressor inlet pressure on the total generated work and cooling

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Compressor pressure ratio analysis

This section examines the influence of the compressor pressure ratio as the total pressure ratio of the Brayton cycle on different parts of the system. An increase in the pressure ratio of the Brayton cycle first increases the specific work consumed by the compressor, as shown in Fig. 6. In addition, the specific work generated by the LP and HP turbines also increases due to the increase in the pressure ratio of each turbine. On the other hand, a pressure rise increases the compressor outlet temperature, leading to an increase in the temperature of Point 3 in the Brayton cycle. Therefore, since the energy transferred from the inlet exhaust gas to the heat exchanger is constant, the flow rate increases. Finally, the total power generated by the Brayton cycle is highest at a pressure ratio of 5.2 owing to the increase in the power consumption of the compressor and the power generation of the turbines.

Fig. 6
figure 6

The effect of an increase in the pressure ratio on a the specific work of the compressor and HP and LP turbines and the flow rate in the Brayton cycle and b the total power consumption of the compressor and LP and HP turbines and the total work generated by the Brayton cycle

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As mentioned previously, a rise in the compressor pressure ratio leads to an increase in the power generated by the turbines. As a result, the outlet temperature of the HP turbine decreases. This leads to an increase in the energy absorbed by the reheater. Hence, the energy entering the absorption chiller cycle decreases. Figure 7 depicts the impact of an increase in the pressure ratio on the energy absorbed in the reheater and the absorption chiller.

Fig. 7
figure 7

The effect of an increase in the pressure ratio on the energy absorbed in the reheater and the absorption chiller

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The absorption chiller cycle is mostly dependent on the generator’s inlet energy, which determines its generated cooling. The energy entering the absorption chiller comes from the exhaust of the Brayton cycle. Accordingly, the energy inlet to the absorption chiller heat exchanger decreases with an increase in the pressure ratio of the Brayton cycle, as shown before. As a result, the cooling generated by the chiller also decreases. Nevertheless, the COP slightly changes due to its simultaneous dependence on the inlet energy and the generated cooling. Figure 8 shows the effect of an increase in the pressure ratio on the cooling generation and the COP of the absorption chiller.

Fig. 8
figure 8

The effect of an increase in the pressure ratio on the cooling generation and the COP of the absorption chiller

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An increase in the pressure ratio of the Brayton cycle reduces the LP turbine outlet temperature. On the other hand, the flow rate in the Brayton cycle increases, as previously stated. Finally, the energy entering the evaporator of the ORC increases, leading to an increase in the power generated by this cycle. Figure 9 shows the effect of the overall pressure ratio of the Brayton cycle on the inlet energy and power generation of the ORC.

Fig. 9
figure 9

The effect of the overall pressure ratio of the Brayton cycle on the inlet energy and power generation of the ORC

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Finally, the effects of changes in the pressure ratio on the overall energy and exergy efficiencies of the system were studied, and the results are shown in Fig. 10. As demonstrated previously, the power generated by the Brayton cycle was maximum at a pressure ratio of 5.2. On the other hand, the power generation in the ORC increases continuously with an increase in the pressure ratio. Moreover, the total power generated by the system is maximum at a pressure ratio of 5.4. On the other hand, the energy efficiency of the system has a decreasing trend due to the decrease in the cooling generated by the absorption chiller cycle. However, the exergy efficiency of the system is maximum at a pressure ratio of 4.73. The reason for the difference of this point from the point of maximum generated power is the decreasing trend in the exergy efficiency of the absorption chiller.

Fig. 10
figure 10

Effect of the overall Brayton pressure ratio on the total power generation and the energy and exergy efficiencies of the system

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Exergy is a thermodynamic term and refers to the maximum amount of useful work that can be obtained from a system in the process of achieving thermodynamic exchange. Exergy also means accessibility. Availability expresses the most work that can be extracted from the system. On the other hand, internal energy expresses the kinetic energy and potential of a micro-scale system. For example, on the scale of atoms and molecules, this energy can be described using their vibrational, rotational, and translational energies. Exergy can be defined as the amount of work that can be extracted from a system that has reached its thermodynamic equilibrium state through a reversible process. Also, there is a certain relationship between the entropy produced by a system and exergy. The production of entropy is equal to the amount of exergy lost. Entropy shows the inability of the system to produce work and exergy shows the ability of the system to produce work.

Finally, the optimal point of the system corresponding to the mentioned temperature conditions and working fluid was determined using the single-objective Pareto method and the TOPSIS technique based on power generation, energy efficiency, and exergy efficiency. Figure 11 shows the Pareto solution and the selection of the optimal point via the TOPSIS method. The optimal pressure ratio in the TOPSIS solution is 4.942.

Fig. 11
figure 11

The Pareto solution and the selection of the optimal point via the TOPSIS method

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Determination of the turbine's pressure ratio in the Brayton cycle

In the previous section, the compressor pressure ratio, which is equal to the overall pressure ratio of the system, was examined using energy and exergy analysis, and its optimum value was determined. On the other hand, given that the system has two turbines in the Brayton cycle, the pressure ratio of each turbine can considerably affect the overall performance of the system. Accordingly, this section conducts an energy and exergy analysis in order to optimize the pressure ratio between the two turbines. To this end, first, the intermediation pressure ratio (IPR) parameter, which is obtained by dividing the pressure ratio of the HP turbine by that of the LP turbine, is introduced, as shown in Eq. (11).

$${\text{IPR}} = \frac{{{\text{PR}}_{{{\text{HP}}}} }}{{{\text{PR}}_{{{\text{LP}}}} }}$$
(12)

In this equation, PRHP is the pressure ratio of the HP turbine, while PRLP is that of the LP turbine. As mentioned previously, the compressor pressure ratio is fixed, and only the pressure ratios of the LP and HP turbines vary. As a result, the compressor power consumption remains constant. As seen before, an increase in the IPR indicates a rise in the pressure ratio of the HP turbine. Figure 12 displays the effect of an increase in the IPR on the power generated by the LP and HP turbines and the total power generated by the Brayton cycle. As seen in the figure, increasing the IPC increases the HP turbine power generation and decreases the LP turbine power generation. This is due to the difference between the inlet and outlet enthalpies of the turbines. Moreover, the total power generated in the Brayton cycle has a maximum point considering the opposite power generation trends in the two turbines and the constant compressor power consumption. The maximum IPC is equal to 1.591.

Fig. 12
figure 12

The effect of an increase in the IPR on the power generated by the LP and HP turbines and the total power generated by the Brayton cycle

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In the next step, the effect of the IPC on the cooling generated by the absorption chiller and the power generated by the ORC is studied. Figure 13 depicts the effect of an increase in the IPR on the power generation of the ORC and cooling of the absorption chiller. As can be seen, the generated cooling is reduced with a rise in the IPR. This is due to the fact that an increase in the IPR reduces the HP turbine outlet temperature, hence increasing heat absorption in the reheater. This, in turn, decreases the heat inlet to the absorption chiller, resulting in a fall in the generated cooling. On the other hand, the outlet temperature of the LP turbine increases due to a decrease in its pressure ratio. This results in an increase in the heat entering the ORC and a rise in its generated power.

Fig. 13
figure 13

The effect of an increase in the IPR on the power generation of the ORC and cooling of the absorption chiller

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Finally, the total power generated by the system and its energy and exergy efficiencies are studied to determine the optimal value of the IPR. Figure 14 depicts the effect of the IPC on the total power generation and the exergy and energy efficiencies of the system. As shown in the figure, the total power is maximum at IPR = 1.918. Also, the exergy efficiency is maximum at IPR = 1.247.

Fig. 14
figure 14

The effect of the IPC on the total power generation and the exergy and energy efficiencies of the system

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The energy and exergy studies indicated that the point of maximum exergy efficiency does not coincide with the point of maximum power generation. Accordingly, the Pareto solution was used to select the appropriate IPR value. In addition, the TOPSIS technique was employed to select the optimal point. Figure 15 shows the Pareto solution and the selection of the optimal point based on the TOPSIS technique. As a result, the optimal IPR value for the mentioned conditions was obtained to be 1.389. Therefore, the pressure ratios of the HP and LP turbines are, respectively, equal to 2.563 and 1.845.

Fig. 15
figure 15

The Pareto solution and the selection of the optimal point via the TOPSIS method

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The results showed that in the temperature range of 300 to 800 0C, helium gas is the most optimal option compared to other gases, because it has the highest amount of electricity production. For this reason, the optimization process is carried out for the best gas that is identified as helium.

Optimization

The previous section involved a study of the parameters affecting the system, namely the Brayton cycle initial pressure, the compressor pressure ratio, and the IPR. The results showed that changes in the parameters affect the various parts of the system differently. Thus, the Pareto solution was used for the optimization and the selection of suitable values. To this end, the compressor pressure ratio and the IPR were considered the variables, and the total power generation and the energy and exergy efficiencies were considered the objective functions. Next, the optimal values in the Pareto solution were calculated using the TOPSIS method. Table 6 presents the system parameters based on the optimal values obtained from the Pareto solution. The optimization results are shown for a temperature range of 300–800 °C. Based on the results, He gas led to the highest power generation at various temperatures. Therefore, if closed Brayton cycles can be used with He as the working fluid, considerably more power can be generated compared to the case of using air.

Table 6 System optimization results for different heat sources and the suitable working fluid
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Conclusions

With the increase in world population and limited energy resources, countries have faced the problem of energy consumption. Crises that threaten countries and human societies, for this reason, methods that can maximize the performance of energy production systems have been the focus of researchers. The present research examined the use of waste heat in different industries for operating a hybrid closed Brayton cycle, ORC, and absorption chiller system. The temperature range used in this research covered all the heat sources and waste from 800 to 300 °C. In addition, HP and LP turbines were used to improve the system performance. The working fluids examined in this research were air, helium, oxygen, carbon dioxide, xenon, and nitrogen. The parameters affecting the performance of the system, namely the compressor inlet pressure, the compressor pressure ratio, and the IPR were investigated. The results indicated that an increase in the compressor pressure ratio does not significantly affect the system performance; hence, the inlet pressure was considered equal to 100 kPa. Moreover, the parameter study showed that changes in different parameters affect the various parts of the system differently, i.e., positively in some cases and adversely in others. As a result, the Pareto solution was used for optimization, and the TOPSIS technique was employed to obtain the optimal values. Based on the optimization results, using helium leads to a higher increase in power generation compared to other gases.

The summary of the results of this research can be stated as follows:

  • Increasing the pressure of the compressor does not have a significant effect on the electricity consumption and the temperature of the working fluid due to the constant pressure ratio.

  • Increasing the Brayton cycle pressure ratio first increases the work consumed by the compressor.

  • The power produced by the Brayton cycle at a pressure ratio of 5.2 is the highest due to the increase in compressor power consumption and turbine power generation.

  • The input energy to the heat exchanger of the absorption chiller decreases with the increase in the Brayton cycle pressure ratio, and as a result, the cooling created by the chiller also decreases.

  • The results of the parametric study showed that the energy efficiency of the system is decreasing due to the reduction of the cooling produced by the absorption chiller cycle.

  • The results of the parametric study showed that the exergy efficiency of the system has the maximum value at the pressure ratio of 4.73.

  • Compressor pressure ratio and intermediation pressure ratio were considered as variables and total power production and energy efficiency and exergy were considered as objective functions.

  • The optimization results were shown for the temperature range of 300–800 °C.

  • The most optimal value of intermediation pressure ratio (IPR) was obtained after the optimization process of 1.389. Also, the most optimal value of pressure ratio of HP and LP turbines was reported as 2.563 and 1.845, respectively.