Introduction

Using \({{\text{CO}}}_{2}\) as a refrigerant has the benefit of low global warming potential (GWP). Another advantage of \({{\text{CO}}}_{2}\) refrigerant is its ozone depletion potential (ODP) which is zero and it has no negative impact on the ozone layer. Striking features of \({{\text{CO}}}_{2}\) refrigerant such as non-flammability, inexpensiveness, and availability increases its practical application in industry [1, 2]. Hydrofluorocarbons such as R134a are detrimental to environment due to their high GWP index in contrast to \({{\text{CO}}}_{2}\) refrigerant [3]. To increase the performance of refrigeration systems, numerous strategies such as applying an expander [4, 5], vortex tube [6], mechanical sub-cooling [7, 8], and cascade \(C{O}_{2}-N{H}_{3}\) refrigeration cycle [9, 10] has been implemented. Due to distinguishing features such as simple geometry, and lack of intricate parts, the ejector refrigeration systems received the attention of designers [11, 12]. Ejector could assist the compressor and decreases compressor work accordingly [13]. The ejector increases the inlet pressure of the following compressor, consequently, it reduces the compressor input work [14]. The ejector could improve the flow rate, evaporator heat exchange, and the exergy efficiency [15].

Wang et al. [16] studied a trans-critical \({{\text{CO}}}_{2}\) ejector heat pump. Their study evaluated the \({{\text{CO}}}_{2}\) heat pump system which includes an ejector device. In their study, the system was investigated from exergy and sensitivity analyses viewpoints. According to their results, the COP of the cycle with ejector and internal heat exchanger increased by 7.38% at 15 °C ambient temperature and the pressure ratio reduced to 2.66. Their results showed the exergy destruction rate of throttling valve was decreased when ejector was applied to the heat pump system.

Casi et al. [17] evaluated the performance of a \({{\text{CO}}}_{2}\) refrigeration plant using a thermo-electric subcooler experimentally. Their study utilizes internal heat exchanger and thermo-electric subcooler together. The experiments have been conducted at different ambient temperature and ambient relative humidity. The optimum pressure of the gas cooler was 71 bar and optimum voltage of thermo-electric devices was 2 V. The experiments reports stated that 22.4% increase in COP was achieved at 35 °C ambient temperature and 75% relative humidity.

Eskandari and Cheraghi [13] investigated a refrigeration cycle with two ejectors. The ejector was modeled using a one-dimensional method. Their proposed new system includes two ejectors and two separators. According to the results, the power consumption for compressor decreased and as a result the gas cooler waste heat decreased. The optimum gas cooler working pressure was 10,000 kPa. The results demonstrated that the exergy destruction rates for each of the components in the proposed cycle decreased in comparison to conventional ejector cycle.

Liu et al. [12] performed a study on a system using two evaporators and two ejectors for supermarkets application. Their cycle utilizes a flash tank. The first ejector increases the main flow of the other one and consequently lift the inlet pressure of the compressor. According to the results, up to 19.1% reduction in pressure ratio was obtained, and COP could improve to 27.1% in comparison to the conventional cycle which includes two evaporators. The optimum pressure of gas cooler for the proposed cycle was around 8.15 MPa.

Santini et al. [18] conducted an experimental and theoretical investigation on trans-critical \({{\text{CO}}}_{2}\) cycle. The experimental system utilizes one compressor and it doesn’t use ejectors. According to their reports, there was a considerable exergy destruction in high pressure expansion valve which was 27.19% of total irreversibility in the experimental operating cycle. In order to overcome this drawback, the expansion valve was replaced by an ejector in a theoretical thermodynamic model. It was obtained that COP is highly dependent on the entrainment ratio and higher COP could be achieved as entrainment ratio rises higher than 0.6. The ejector assisted the cycle in decreasing the temperature of compressor outlet and also it decreased the power consumption of compressor.

Aranguren et al. [19] conducted an experimental test of trans-critical single stage \({{\text{CO}}}_{2}\) refrigeration plant including a thermo-electric subcooler. The tests are done on experimental system under relative humidity of 55%. The thermo-electric was placed at the outflow of the gas cooler and they were in series configuration. The capacity of cooling was about 280 W. The applied thermo-electric had four copper blocks. The results demonstrated the optimum COP was 1.15 for the overall system. At this optimal working condition there was a 15.3% rise in cooling capacity compared to the cycle without thermo-electric subcooler.

In this paper, the ejector two-stage trans critical refrigeration cycle is analyzed from energy, exergy, and exergo-economic viewpoints. The cycle was modeled by Engineering Equation Solver (EES). A parametric study has been performed to evaluate the effects of gas cooler pressure, evaporator pressure (saturation temperature), and low-pressure compressor discharge pressure on the COP. Energy, exergy, and exergo-economic analyses were conducted to determine the inefficiencies in cycle operation. In addition, a multi-objective optimization using genetic algorithm, has been conducted to achieve a balance between the coefficient of performance and refrigeration cost rate. A thermo-electric generator (TEG) was applied for converting waste heat to electricity. Studying components under different operating conditions such as different operating pressures is essential to enhance the efficiency of the refrigeration cycle [20]. Applying a thermo-electric generator is a method to utilize waste heat from an industrial component, it also has no moving parts and has a low maintenance cost [21].

System properties and assumptions

The components and property diagram of the introduced cycle are depicted in Figs. 1 and 2, respectively. The \({{\text{CO}}}_{2}\) refrigerant first comes from the intercooler in the saturated vapor phase (state 1) then the high-pressure compressor is conducting a pressure raise and the refrigerant reaches a high pressure (state 2). The gas then transfers heat to the cooling water (state 3) and then is used for preheating the evaporator output flow, through the heat exchanger (state 4). Then it enters the primary nozzle and is mixed in the mixing chamber with the outflow of the low-pressure compressor, and finally passes the diffuser to be pressurized (state 5). In the intercooler, input flows which are ejector output (state 5) and medium pressure expansion valve output (state 11), exchange heat. Then saturated vapor and saturated liquid flow and enter the compressor and the expansion valve, respectively (state 1 and state 6).

Fig. 1
figure 1

Schematic diagram of the refrigeration cycle

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Fig. 2
figure 2

Phase diagram of the studied refrigeration cycle

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After throttling by low-pressure expansion valve (state 7), the refrigerant enters the evaporator and performs cooling and exchanging heat with the air stream. After the evaporator (state 8), refrigerant passes through the internal heat exchanger and enters the compressor (state 9). Then one part of the refrigerant enters the ejector and the other part is throttled through the medium-pressure expansion valve and enters the intercooler. Two separate water streams are used to transfer the excess heat from the gas cooler. For simplicity of simulation some assumptions are considered as follows [20]:

  • The adiabatic non-isentropic compression processes occur in the refrigeration cycle.

  • Fluid mixing in the mixing chamber takes place at constant pressure.

  • The pressure and temperature of the atmospheric state are \({P}_{0}=101.325 kPa\) and \({T}_{0}=25^\circ C\) respectively.

  • Pressure drops and heat loss of internal heat exchanger, evaporator, gas cooler, pipes, and intercooler are ignored.

  • The temperature of the inlet water and outlet water of the gas cooler and inlet air and outlet air of the evaporator and inlet water and outlet of the thermo-electric generator cold side (state 12) remain fixed during the cycle.

  • The kinetic energy of the inlet flow of the ejector is ignored.

  • The state points of flowing air through the evaporator (states 16 and 17) are fixed, i.e., refrigeration capacity is constant [20].

  • The inlet and output states of internal heat exchanger, are remained fixed (Fig. 3).

Fig. 3
figure 3

Schematic diagram of two-phase ejector

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Energy and exergy analysis

Energy model

According to previous assumptions, the energy model was obtained using mass and energy conservations.

For compressors:

$${W}_\text{lpc}={\dot{m}}_\text{lpc}({h}_{10}-{h}_{9})$$
(1)
$${W}_\text{hpc}={\dot{m}}_\text{hpc}\left({h}_{2}-{h}_{1}\right)$$
(2)
$${\eta }_\text{lpc}=\frac{{h}_{10.is}-{h}_{9}}{{h}_{10}-{h}_{9}}$$
(3)
$${\eta }_\text{hpc}=\frac{{h}_{2.is}-{h}_{1}}{{h}_{2}-{h}_{1}}$$
(4)

where \({W}_\text{lpc}\), \({W}_\text{hpc}\), and \({h}_\text{is}\) are the low-pressure compressor input power, high-pressure compressor input power, and isentropic enthalpy of the compressors outlet, respectively. According to Ref. [22], the compressors adiabatic efficiencies are:

$${\eta }_\text{lpc}=0.815+0.022\left(\frac{{P}_{10}}{{P}_{9}}\right)-0.0041{\left(\frac{{P}_{10}}{{P}_{9}}\right)}^{2}+0.0001{\left(\frac{{P}_{10}}{{P}_{9}}\right)}^{3}$$
(5)
$${\eta }_\text{hpc}=0.815+0.022\left(\frac{{P}_{2}}{{P}_{1}}\right)-0.0041{\left(\frac{{P}_{2}}{{P}_{1}}\right)}^{2}+0.0001{\left(\frac{{P}_{2}}{{P}_{1}}\right)}^{3}$$
(6)

where \(\eta\) is the adiabatic efficiency and could be calculated using the inlet and outlet pressure of each compressor. For evaporator refrigeration capacity:

$${\dot{Q}}_\text{evaporator}={\dot{m}}_\text{low}({h}_{8}-{h}_{7})$$
(7)

where \({\dot{Q}}_\text{evaporator}\) is the heat transfer rate in evaporator. The outflow of the internal exchanger enters the ejector primary inlet and the outflow of the compressor enters the ejector’s secondary inlet. Finally, after mixing process, the mixture is expanded and flows toward the intercooler. The governing equations for ejector are as follows [23]. Using the first law of thermodynamics, the velocity out of the nozzle can be derived as:

$${v}_\text{p}=\sqrt{2{\eta }_\text{n}({h}_\text{n.in}-{h}_\text{n.out.is})\times 1000}$$
(8)

where \(h_{{n.{\text{in}}}}\) is the specific enthalpy of the ejector’s primary inlet and \(h_{{n.{\text{out}}.{\text{is}}}}\) is the exit isentropic specific enthalpy in the nozzle with constant pressure which is equal to low-pressure compressor discharge. Considering momentum conservation, velocity of the mixture in mixing chamber \({v}_\text{m}\) can be derived as:

$${v}_\text{m}=\frac{{v}_\text{p}}{1+\omega }\sqrt{{\eta }_\text{m}}$$
(9)

where \(\omega\) is the entrainment ratio of the ejector. Applying the first law of thermodynamics, the specific enthalpy of mixed refrigerant \({h}_\text{m}\) is calculated as:

$${h}_\text{m}=\frac{{h}_\text{n.in}+\omega {h}_\text{sec.in}}{1+\omega }-\frac{{v}_\text{m}^{2}}{2\times 1000}$$
(10)

where \({h}_{{\text{sec}}.{\text{in}}}\) in the specific enthalpy of the secondary inlet.

By applying the diffuser efficiency, the enthalpy at the ejector outlet could be derived as:

$${h}_\text{d}={h}_\text{m}+\frac{{h}_\text{d.out.is}-{h}_\text{m}}{{\eta }_\text{d}}$$
(11)

where \({h}_\text{d.out.is}\) is the diffuser exit isentropic specific enthalpy.

Using energy conservation, the velocity at the outlet of the ejector is derived as:

$${v}_\text{d}=\sqrt{{v}_\text{m}^{2}-2({h}_\text{d}-{h}_\text{m})\times 1000}$$
(12)

The entrainment ratio is obtained by an iteration based on Fig. 4. Applying a mass balance through the two-phase ejector, the quality \({x}_\text{do}\) is derived as a function of entrainment ratio i.e.:

Fig. 4
figure 4

Flowchart of ejector entrainment ratio calculations

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$${x}_\text{do}=\frac{1}{1+\omega }$$
(13)

A tolerance of 0.01 is used for convergence. Applying the first law of thermodynamics to ejector:

$$h_{4} + \omega h_{{10}} = \left( {1 + \omega } \right) \cdot \left( {h_{5} + \frac{{v_{{\text{d}}} ^{2} }}{{2000}}} \right)$$
(14)

The assumptions for the iterative process to calculate entrainment ratio are pressures and temperatures of primary and secondary inlets, the outlet pressure of ejectors, and mixing chamber, diffuser, and nozzle efficiencies. These parameters are used to calculate the velocities and enthalpies of the ejector different parts. Finally, the calculated entrainment ratio is used to calculate the mass flow rate of high-pressure stage of the cycle (Table 1).

Table 1 Parameters and assumptions for ejector [20]
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For thermo-electric generators according to Ref. [24] the equations are as follows:

$${Q}_{{\text{h}}.{\text{TEG}}}={\dot{m}}_{{\text{water}}}({h}_{14}-{h}_{13})$$
(15)

where \({Q}_{{\text{h}}.{\text{TEG}}}\) is the heat received from gas cooler water.

$${P}_{{\text{TEG}}}={Q}_{{\text{h}}.{\text{TEG}}}\times {\eta }_{{\text{TEG}}}$$
(16)

where \({P}_\text{TEG}\) is the generated power of the thermo-electric generator.

The thermo-electric generator efficiency \(({\eta }_\text{TEG})\) is defined as a function of Carnot efficiency \({(\eta }_\text{Carnot})\), working temperatures, and material properties.

$${\eta }_\text{TEG}={\eta }_\text{Carnot}\times \frac{\sqrt{1+{ZT}_\text{average}}-1}{\sqrt{1+{ZT}_\text{average}}+\frac{{T}_\text{c}}{{T}_\text{h}}}$$
(17)
$${\eta }_\text{Carnot}=\frac{{T}_\text{h}-{T}_\text{c}}{{T}_\text{h}}$$
(18)

\({ZT}_\text{average}\) is the non-dimensional figure of merit of the semiconductor and is defined as:

$${ZT}_\text{average}=\frac{{\alpha }^{2}\times \sigma }{\lambda }\times \frac{({T}_\text{h}+{T}_\text{c})}{2}$$
(19)

where \({T}_\text{h}\), \({T}_\text{c}\), and \({T}_\text{average}\) are the average temperature of hot side, the average temperature of the cold side, and the average operating temperature of the thermo-electric generator, respectively. The properties of thermo-electric are listed in Table 2.

Table 2 Thermo-electric material properties [24]
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For calculating the high-stage refrigerant mass flow rate, the following mass balance at low-pressure compressor node is written:

$$\dot{m}_\text{low} = \omega \dot{m}_\text{high} + \dot{m}_{11}$$
(20)

Which \({\dot{m}}_\text{low}\) could be calculated using energy balance across the evaporator. The COP of the system was calculated as follows:

$$COP=\frac{{Q}_\text{e}}{{W}_\text{lpc}+{W}_\text{hpc}-{P}_\text{TEG}}$$
(21)

Exergy analysis

The ability of a system to produce work during a set of processes to reach equilibrium with the environment is defined as exergy. Conventional exergy analysis (CEA) is a usual method for measuring irreversibility of components [25]. The CEA, can determine the position and size of the exergy destruction in system components [26]. The Flow exergy of each state point was calculated as [20]:

$${\dot{E}}_\text{j}=\dot{m} \cdot {e}_\text{j}=\dot{m} \cdot \left[\left({h}_\text{j}-{h}_{0}\right)-{T}_{0}({s}_\text{j}-{s}_{0})\right]$$
(22)

where subscript j indicates the 1–17 state points in the cycle property diagram (Fig. 2) and subscript 0 indicates the reference state point. For kth component of the system, the conventional exergy balance can be written as:

$${\dot{E}}_\text{F.k}={\dot{E}}_\text{P.k}+{\dot{E}}_\text{D.k}$$
(23)

The flow exergy of each line can be separated into thermal and mechanical exergy i.e.:

$$e = e_{{{\text{j}}^{T} }} + e_{{{\text{j}}^{M} }} = \left[ {\left( {h_{{{\text{j}} - {\text{h}}\;{\text{P}}_{{\text{j}}} {\text{.T}}_{{\text{0}}} }} } \right) - T_{0} \left( {s_{{\text{j}}} - s_{{{\text{P}}_{{\text{j}}} .{\text{T}}_{{\text{0}}} }} } \right)} \right]_{{{\text{P}} = {\text{const}}}} + \left[ {\left( {h_{{{\text{P}}_{{\text{j}}} .{\text{T}}_{{\text{0}}} }} - h_{0} } \right) - T_{0} \left( {s_{{{\text{P}}_{{\text{j}}} .{\text{T}}_{{\text{0}}} }} - s_{0} } \right)} \right]_{{{\text{T}} = {\text{const}}}}$$
(24)

The total exergy balance is:

$${\dot{E}}_\text{F.total}={\dot{E}}_\text{P.total}+\sum {\dot{E}}_\text{D.k}+{\dot{E}}_\text{L.total}$$
(25)

\({\dot{E}}_\text{L.total}\) is the exergy that cannot be used in the system and can be calculated using Eq. (22). In the CEA method, to assess component and overall system performance, the following parameters are defined:

$${\eta }_\text{k}=\frac{{\dot{E}}_\text{P.k}}{{\dot{E}}_\text{F.k}}$$
(26)
$${\eta }_\text{ex}=\frac{{\dot{E}}_\text{P.total}}{{\dot{E}}_\text{F.total}}$$
(27)
$${\delta }_\text{k}=\frac{{\dot{E}}_\text{D.k}}{\sum {\dot{E}}_\text{D.k}}$$
(28)
$${\delta }_\text{ex}=\frac{\sum {\dot{E}}_\text{D.k}}{{\dot{E}}_\text{F.total}}$$
(29)

\({\eta }_\text{k}\) and \({\eta }_\text{ex}\) are the exergy efficiency of each component and the exergy efficiency of the overall system, and \({\delta }_\text{k}\) and \({\delta }_\text{ex}\) are the exergy destruction ratios of each component and overall system, respectively [20] (Table 3, 4, 5 and 6).

Table 3 Fuel and product exergy equation for each system component
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Table 4 Exergo-economic data [27]
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Table 5 The investment costs of components [29,30,31]
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Table 6 Cost balance equation for exergo-economic analysis
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Exergo-economic analysis

Economy factor is an indispensable part in engineering applications. In order to calculate the cost for the exergy of streams, exergo-economic analysis complements the exergy analysis [27]. Exergo-economic analysis is a powerful technique for specifying cost- effectiveness of a thermodynamic system [28]. For each component, the cost balance is written as [27]:

$${\dot{Z}}_\text{k}+{\dot{C}}_\text{q}+\sum {\dot{C}}_\text{i}=\sum {\dot{C}}_\text{e}+{\dot{C}}_\text{w}$$
(30)

where \({\dot{Z}}_\text{k}\), \({\dot{C}}_\text{q}\), \({\dot{C}}_\text{i}\), \({\dot{C}}_\text{e}\), and \({\dot{C}}_\text{w}\) are the total investment, heat transfer, inlet stream, exit stream, and work cost rate for each component in the system.

The ratio of cost rate to exergy rate is defined as specific exergy cost:

$${c}_\text{j}=\frac{{\dot{C}}_\text{j}}{{\dot{E}}_\text{j}}$$
(31)

Total investment cost rate could be calculated using the component’s investment cost as follows [27]:

$${\dot{Z}}_\text{k}=\left(\frac{CRF\times \varphi }{N\times 3600}\right){Z}_\text{k}$$
(32)

where \({\dot{Z}}_\text{k}\) is the total investment cost rate for each component,\(\varphi\) is the system maintenance factor, \(N\) is the system operating hour in the year, and \({Z}_\text{k}\) is the component’s investment cost.

The capital recovery factor is calculated as:

$$CRF=\frac{{i}_\text{r}{\left(1+{i}_\text{r}\right)}^{\text{n}}}{{\left(1+{i}_\text{r}\right)}^{\text{n}}-1}$$
(33)

where \({i}_\text{r}\) is an interest rate that compensates for investment in equipment and \(n\) is the system life.

Optimization method

There are different evolutionary optimization algorithms such as differential evolution algorithms (DE) [33, 34], harmony search algorithm (HS) [35, 36] and genetic algorithm (GA) [37]. The evolutionary algorithms are based on Darwinian evolution which are well-known and are employed to optimize complex engineering problems such as fluid dynamics optimization [38], metallurgical problems [39] and thermodynamic systems [40]. Genetic algorithms are global optimization methods that search for optimums through the imitation of the natural evolution of biological organisms [41]. The optimization was done by NSGA-II (non-dominated sorting genetic algorithm). NSGA-II algorithm optimize the function by keeping elites of each population. Elites are fine results of the previous populations [42]. Multi-objective optimization has been conducted to maximize COP and minimize the refrigeration cost rate using MATLAB genetic algorithm. Multi-objective optimization is used when there is a conflict between the objectives [43]. In refrigeration systems, by increasing COP, the cost rate increases which is a conflict. Indeed, there is a trade-off between COP and cost rate.

Results and discussion

Base cycle result

The results of the base cycle analysis are presented in Table 7 and 8. It was discovered that ejector outlet flow has the highest cost rate (5.393 $ h−1). The refrigeration cost rate (C7) is 2.898 $ h−1. The water mass flow rate of the gas cooler was calculated as 0.1171 kg s−1. The low compressor discharge temperature is 130.5 °C which is the highest temperature in the refrigeration cycle. The COP of the base cycle is calculated as \(1.593\) which has increased by 27.24% in comparison to Ref. [20]. The ejector entrainment ratio was calculated as 0.47. The exergy efficiency and exergy destruction ratio of the cycle, are \({\eta }_\text{ex}=0.3485\), \({\delta }_\text{ex}=0.5081\)

Table 7 Exergy and exergo-economic results for each stream
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Table 8 Exergy results for each component of the cycle
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According to Table 8, expansion valve 1, thermo-electric generator, and low-pressure compressor have the highest exergy destruction rate as 0.9809 kW, 0.8364 kW, and 0.7308 kW respectively. The intercooler, gas cooler, and low-pressure compressor have the highest exergy efficiency which are 0.995, 0.971, and 0.883. Expansion valve 1, thermo-electric generator, and low-pressure compressor have the highest exergy destruction ratio as 0.211, 0.180, and 0.158 respectively; which are the high priorities for system improvement.

Parametric study results

To achieve a comprehensive view about performance of suggested system a parametric study has been conducted on the gas cooler, low-pressure compressor discharge pressure, and evaporator pressure for pressure ranges of 8800–20000 kPa, 4500–5500 kPa, and 800–1200 kPa, respectively (Fig. 5).

Fig. 5
figure 5

a Efficiency and b exergy destruction ratio of each component

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Gas cooler pressure effects

Figure 6a depicts the effect of gas cooler pressure on COP and C7. It was discovered by increasing gas cooler pressure, the COP of the cycle drops from 1.693 to 0.8289. Also, by increasing gas cooler pressure, the refrigeration cost rate (C7) is approximately constant and it is roughly 2.864 $ h−1. According to Fig. 6b, it can be deduced that by increasing gas cooler pressure, total fuel exergy rate, exergy destruction ratio, and total exergy destruction rate, rise steadily but exergy efficiency of the cycle drops from 0.3686 to 0.1899. Figure 6c demonstrated how system components are affected by gas cooler pressure. As pressure rises, exergy destruction of the gas cooler, ejector, and thermo-electric generator, increased. Gas cooler exergy destruction rate increased from 0.2113 kW to 3.243 kW which suggested reducing gas cooler pressure in order to reduce the exergy destruction rate of gas cooler to enhance system operation. The higher pressure of gas cooler could increase the input power of the compressor and also it could increase the outlet temperature of high-pressure compressor. Finally, it increased the exergy destruction of the compressor and gas cooler.

Fig. 6
figure 6

a Effect of gas cooler pressure on refrigeration cost rate and COP b Effect of gas cooler pressure on cycle exergy destruction ratio, cycle efficiency, total exergy destruction, and total fuel exergy c Effect of gas cooler pressure on exergy destruction of components

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Evaporator pressure effects

Figure 7a demonstrated the effect of evaporator pressure on the COP and refrigeration cost rate. As depicted, by the rise of evaporator pressure, COP increased from 1.174 to 1.593 and the refrigeration cost rate decreased from 3.0528 $ h−1 to 2.898 $ h−1. Based on Fig. 7b by the rise of evaporator pressure, total fuel exergy rate, exergy destruction ratio, and total exergy destruction rate, decreased and exergy efficiency of the cycle increased by 34.82% (from 0.2585 to 0.3485). Figure 7c demonstrated the effect of evaporator pressure on exergy destruction of components. As evaporator pressure increased, the exergy destruction of the evaporator, expansion valve 1, and intercooler, decreased and exergy destruction of the ejector increased from 0.1122 kW to 0.5092 kW.

Fig. 7
figure 7

a Effect of evaporator pressure on refrigeration cost rate and COP. b Effect of evaporator pressure on cycle exergy destruction ratio, cycle efficiency, total exergy destruction, and total fuel exergy. c Effect of evaporator pressure on exergy destruction of components

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Discharge pressure of low-pressure compressor effects

According to Fig. 8a by increasing in discharge pressure of low-pressure compressor, COP decreased from 1.615 to 1.432 and the refrigeration cost rate increased from 2.863 $ h−1 to 3.189 $ h−1. Figure 8b depicts the effect of discharge pressure on total fuel exergy rate, exergy destruction ratio, total exergy destruction rate, and exergy efficiency. It can be seen that by increasing discharge pressure, total fuel exergy rate increased; and total exergy destruction rate, exergy destruction ratio and exergy efficiency of the overall system decreased. Figure 8c demonstrated the effect of discharge pressure on exergy destruction of components. As discharge pressure increased, exergy destruction rate of low-pressure compressor, expansion valve 2, and intercooler increased and exergy destruction rate of the ejector decreased due to higher secondary inlet exergy based on exergy balance for ejector.

Fig. 8
figure 8

a Effect of discharge pressure of low-pressure compressor on refrigeration cost rate and COP. b Effect of discharge pressure of low-pressure compressor on cycle exergy destruction ratio, cycle efficiency, total exergy destruction, and total fuel exergy. c Effect of discharge pressure of low-pressure compressor on exergy destruction of components

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Multi-objective optimization results

As depicted in Fig. 9 firstly, the fitness function, number of variables, and the range of each variable was determined. Then the size of population was set. By using fitness function, the value of each individual was calculated. The new population was generated considering crossover and mutation fractions based on Table 9. The convergence criterion was the maximum number of generations. It was 15 for this process. Figure 10 is the Pareto frontier of optimization. Table 10 is the list of parameters for the corresponding points in Pareto frontier. According to the optimization outputs, the optimum design point, could provide the COP of 1.281. The exergy efficiency and the exergy destruction ratio of the optimum design point are 0.284 and 0.574, respectively and the exergy destruction rate of the optimum point is 6.544 kW.

Fig. 9
figure 9

Flowchart of multi-objective optimization

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Table 9 Genetic algorithm options and parameter ranges for optimization
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Fig. 10
figure 10

Multi-objective optimization result (Pareto frontier)

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Table 10 Multi-objective optimization results
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Conclusions

A study on ejector two-stage trans-critical CO2 refrigeration cycle which is equipped with a thermo-electric generator was conducted. The cycle modeled using Engineering Equation Solver (EES) and the optimization was done using MATLAB multi-objective optimization utility. One of the main purposes of this study was to evaluate the effect of thermo-electric generator in the cycle. Thermo-electric generator, recovers energy from waste heat of gas cooler thus it could reduce the inlet work of compressors and consequently increase the COP. The refrigeration system was assessed from energy, exergy, and exergo-economic viewpoints. The China business electricity price [32] was used for the value of compressors specific exergy cost (cw). According to exergy analysis result, low-pressure expansion valve, low-pressure compressor, and evaporator have the highest exergy destruction rate. The coefficient of performance was obtained as 1.593. The ejector entrainment ratio was 0.47. The exergy efficiency and exergy destruction ratio of the system, were 0.3485 and 0.5081, respectively. The results of the simulation show expansion valve 1, thermo-electric generator, and low-pressure compressor have the highest exergy destruction rate as 0.9809 kW, 0.8364 kW, and 0.7308 kW, respectively. The important results were obtained are:

  • \({\delta }_\text{ex}\) Increases by increasing the gas cooler pressure.

  • \({\delta }_\text{ex}\) Decreases by increasing the evaporator pressure.

  • According to optimization outputs, the optimum pressures for low-pressure compressor outlet and evaporator are about 4518.70 kPa and 1197.44 kPa, respectively.

  • Expansion valve 2 acts as a control valve and it transfers the excess part of the low-pressure compressor outflow to the intercooler.

  • The optimum COP of the system was 1.281.

  • The total exergy destruction rate varies from 5.48 kW to 7.916 kW based on optimization results.

  • The exergy efficiency and the exergy destruction ratio of the optimum design point are 0.284 and 0.574.

Further work is needed to assess the impact of different refrigerant on system performance. Another important analysis could be exergo-environmental analysis which could be combined with exergy and exergo-economic analysis to improve the environmental impact of the system. Furthermore, a renewable energy source could be used to supply the cycle. Solar photovoltaic panel could be employed to produce power and supply the compressors.