Abstract
Cryogenics has an important influence on industry and science. In this study, optimum working conditions are obtained by applying exergy analysis and local optimization methods to two- and three-stage vapor compression cascade cryogenic cycle. The first and second laws of thermodynamics, exergy analysis, and local optimization methods are applied to the two- and three-stage cascade cryogenic cycle. By considering the needs and demands, it is possible to create new cycles by adding new devices and/or new stages to these cycles. The results of the optimum operating conditions are obtained for the two- and three-stage vapor compression cascade cryogenic cycle. It is seen that to achieve high COP values and high efficiency; it is necessary to reduce the compression ratio of the compressor as much as the fluid allows. For the two-stage cycle, the minimum total work required for cryogenic cooling is around P 7 = 2,400 kPa. The COP value is 0.30 between P 7 = 2,400 and 2,800 kPa, and the maximum exergy efficiency is obtained around 0.235. It is seen operating the first-stage compressor at high pressures increases the total losses of the entire cycle from 7,500 to 18,550 kW. The increase in total exergy losses is around 247%, and operating the first-stage compressor at high pressures increases the exergy efficiency of the entire cycle. The increase in total exergy efficiency is around 160%. When the second-stage compressor is operated at low pressure, the COP value increases by 2%, the exergy efficiency increases by 20%, and the exergy losses decrease by around 40%.
1 Introduction
Cryogeny is the term used for processes performed at very low temperatures. The working area of cryogenic processes occurs at temperatures lower than 120 K. Large-scale air separation units use the cryogenic method to separate the elements in the air for use in medicine and industry. The products resulting from the separation process are generally kept in liquid form at cryogenic temperatures for transport and storage. Liquid helium is commonly used to cool superconducting magnets that have been used for MRI (Magnetic Resonance Imaging) systems. In space technology, cryogens such as liquid hydrogen and oxygen have been found and used as fuel in rocket engines and applications. The application of cryogenics can be roughly divided into five main areas: liquefaction and separation of gases, storage and transportation of gases, biological and medical applications, superconductivity, and changing the material and fluid properties at low temperatures. The cryogenic liquid is both the working and cooling agent of a cryogenic system. Only liquids with a triple point below 100 K are considered “cryogenic,” meaning they are still in liquid or gaseous form below this temperature [1,2,3]. Hydrogen is rarely used as a cryogenic coolant due to the instability of liquid hydrogen, and oxygen is never used due to serious hazards. Despite being an inert gas, Neon is very expensive and therefore little used. As a result, the most important cryogenic fluids used are helium and nitrogen. The wide usage area of cryogenic cooling systems emphasizes the importance of air separation method. Nitrogen, oxygen, and argon, which are three different elements of high purity obtained after the air separation method, are used in different applications in the gaseous state, apart from the liquid state. These cryogens are generally obtained in high purity. It is known to be used in the chemical industry and medical applications [4,5,6,7]. Van der Ham and Kjelstrup (2010) evaluated two process designs of cryogenic air separation units by applying exergy analysis. The air separation unit is part of IGCC (integrated gasification combined cycle). In the two processes, similar feed products were divided into products with the same characteristics. In the study, exergy analysis of double- and triple-column designs was evaluated. As a result, the triple column performed better than the double column. While there is a high- and low-pressure column in the double column, there is also a medium-pressure column in the triple column [8]. Thomas et al. worked on the exergy analysis of the liquefaction system of claude-based Helium. While making this analysis, Aspen HYSYS simulation program was used. Claude modified his cycle using two turbines. As a result, he observed the efficiency of the first and second turbines as 75 and 70% [9]. As a result of the readings and literature reviews, it has been revealed that the descending temperature in cooling determines the Cooling Effect Coefficient (COP) value. In Table 1, the COP values obtained in cooling decrease as the cooling temperature decreases. The compressor power required for low cooling temperatures increases and the amount of heat absorbed decreases. It is seen that the Carnot COP value is about three times the actual COP values. This situation shows us that we can approach the Carnot COP value by working on suitable refrigerant selection, improved design, and optimum operating conditions [10,11,12].
Carnot and actual COP values obtained in refrigeration
| T cold | W Comp/Q Heat | Carnot COP | Actual COP |
|---|---|---|---|
| 270 | 0.11 | 9 | 3.33–2.0 |
| 100 | 2 | 0.5 | 0.1–0.05 |
| 20 | 14 | 0.0714 | 0.01–0.005 |
| 4 | 74 | 0.0135 | 0.0014–0.0007 |
2 Materials and methods
The design of a cooling system depends on the thermodynamic properties of the refrigerant considered to be used in the system. Due to the selected refrigerant feature, it is preferred that the evaporator pressure in the system is high and the condenser pressure is low. Fluids that are used to transfer heat from one environment to another in the conversion process make their heat exchanges by converting them from the liquid phase to the vapor phase and from the vapor phase to the liquid phase. Thus, the fluids perform heat exchange through evaporation and condensation processes. While choosing the refrigerant, the damage to the environment should be taken into consideration [13,14,15].
In the vapor compression cooling system, the refrigerant, which is compressed to high pressure in the compressor, is sent to the condenser as superheated steam. The refrigerant performs the condensation process by giving heat to the outside of the condenser. After performing the condensation process, it enters the throttling valve with the help of capillary pipes. The temperature and pressure of the refrigerant in the throttling valve drops significantly and enters the evaporator. It performs the evaporation process by taking the heat of the environment in the evaporator. After performing the evaporation process in the evaporator, it re-enters the compressor and the cycle is completed [4,5].
The obtained highest work rate when the material flow is transformed into the environment determined as P 0 and T 0 from the initial states with physical processes that only thermally interact with the environment is called physical exergy [6]. Availability or exergy is the theoretical maximum amount of useful work that can be obtained if equilibrium with the environment is achieved at the end of a reversible process. It has two components, physical and chemical. The physical exergy of perfect gas mixtures can be written as follows:
While determining the physical exergy, kinetic and potential exergy are defined as zero. The last state is the state that is reduced to the dead state, which is not limited to the environment, as stated in the definition. Since chemical reactions do not occur between the reference materials, only the existence of a complete thermodynamic equilibrium is mentioned and the total exergy of the environment is zero [4,5,6].
The classical cascade system is generally used in the first established liquefaction plants. But when compared to other systems, their price is higher; however, it is widely used in many countries around the world. In the classical cascade system, the cooling process is carried out by using different fluids at each stage. Each refrigerant operates in a closed circuit independent of the other, in a single stage or several stages, in the appropriate pressure and temperature range. The classical cascade system is a system operated in separate closed circuits using different refrigerants. A mixed fluid cascade cooling system is widely used. Although the working principle is similar to the classical cascade system, the number of closed circuits used in the classical cascade system and the high number of compressors used have an important share in terms of system costs, and alternative ways have been developed to reduce these costs in these systems. Here, although the gradual cooling process is performed, it is essential to use the refrigerants in a mixed state [12,13,16].
In this study, the two-stage and three-stage vapor compression cascade cryogenic cycles are taken from the book of Wiley publications named “Refrigeration Systems and Application” by Dinçer and Kanoglu [4]. The cycle in Figure 1, which we are working with, takes place in two stages. In the second stage, the refrigerant enters the compressor as ethane gas. Here, the refrigerant is compressed up to the condenser pressure. While the refrigerant coming out of the compressor at a high temperature passes through the pipes of the condenser, it starts to cool by giving heat to the environment and condensation takes place. The fluid enters a capillary tube after the condenser, and its pressure and temperature drop significantly with the effect of the throttling valve. The low-temperature refrigerant then enters the evaporator, where evaporation takes place by taking heat from the cooled environment. The cycle is completed when the refrigerant leaves the evaporator and then re-enters the compressor. In the first subcycle, the refrigerant enters the compressor as methane gas. The refrigerant is compressed to condenser pressure. While the refrigerant coming out of the compressor at a high temperature passes through the pipes of the condenser, it starts to cool by giving heat to the environment and condensation takes place. The fluid enters a capillary tube after the condenser, and its pressure and temperature drop significantly with the effect of the throttling valve. The low-temperature refrigerant then enters the evaporator, where evaporation takes place by taking heat from the cooled environment. The cycle is completed when the refrigerant leaves the evaporator and then re-enters the compressor [2].
Two-stage vapor compression cascade cryogenic cycle.
In the second subcycle, natural gas enters the system. Here, after the refrigerant condenses, it enters the evaporator and evaporates with the heat it receives from the cooled environment. After the refrigerant leaves the evaporator, it completes the cycle in the form of a liquid natural gas mixture [4,5].
Values of enthalpy, entropy, and other thermodynamic properties in Thermodynamic analysis of cycles are taken from https://webbook.nist.gov/chemistry/fluid/3 website [17].
Also, some other techniques have been applied for different purposes in literature and important results have been obtained [18,19,20,21,22,23,24,25,26,27,28,29,30,31].
In Table 2, the formulas for the mass and energy balance and entropy generation of the two-stage vapor compression cascade cryogenic cycle devices are given.
Formulas for the mass and energy balance and entropy generation of the devices of the two-stage vapor compression cascade cryogenic cycle [2,4,5,6]
| Device | Mass balance | Energy balance | Entropy generation |
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| Compressor1 |
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| Condenser1 |
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| Throttle Valve1 |
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| Evaporator1 |
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| Compressor2 |
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| Condenser2 |
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| Throttle Valve2 |
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| Evaporator2 |
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In Table 3, the formulas of entropy balance, exergy loss, and exergy efficiency of the devices of the two-stage vapor compression cascade cryogenic cycle. The three-stage vapor compression cascade cryogenic cycle is given in Figure 2.
General formulas of entropy balance, exergy loss, and exergy efficiency of the devices of the two-stage vapor compression cascade cryogenic cycle [2,4,5,6]
| Device | Entropy balance | Exergy loss | Exergy efficiency |
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| Compressor |
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| Evaporator |
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Taken from Enviro |
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| Condenser |
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Three-stage vapor compression cascade cryogenic cycle.
In Table 4, the formulas for the mass and energy balance and entropy generation of the three-stage vapor compression cascade cryogenic cycle devices are given.
Formulas for the mass and energy balance and entropy generation of the devices of the three-stage vapor compression cascade cryogenic cycle [2,4,5,6]
| Device | Mass balance | Energy balance | Entropy generation |
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| Compressor1 |
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| Condenser1 |
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| Throttle Valve1 |
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| Evaporator1 |
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| Compressor2 |
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| Condenser2 |
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| Throttle Valve2 |
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| Evaporator2 |
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| Compressor3 |
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In Table 5, the formulas of entropy balance, exergy loss, and exergy efficiency of the devices of the three-stage vapor compression cascade cryogenic cycle are given.
General formulas of entropy balance, exergy loss, and exergy efficiency of the devices of the three-stage vapor compression cascade cryogenic cycle [2,4,5,6]
| Device | Entropy balance | Exergy loss | Exergy efficiency |
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| Compressor |
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| Evaporator |
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Taken from Enviro |
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| Taken from water | |||
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| Condenser |
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| Throttle valve |
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3 Results and discussion
3.1 Two-stage vapor compression cascade cryogenic cycle
Normal conditions (25°C, 100 kPa) are chosen as the reference state. Thus, for water h 0 = 104.92 kJ/kg and s 0 = 0.3672 kJ/kg K, for propane h 0 = 630.43 kJ/kg and s 0 = 2.8474 kJ/kg K, for ethane h 0 = 667 kJ/kg and s 0 = 3.3984 kJ/kg K and for methane h 0 = 909.97 kJ/kg and s 0 = 6.6813 kJ/kg K are taken.
Some of the pressure, temperature, mass flow, enthalpy, entropy, energy, and exergy values of each flow line obtained as a result of the calculations are given in tables. By looking at the energy balance of each case, it was checked whether the calculations were correct and given in the tables. COP values, compressor work (W C), heat energy dissipated in the condenser (Q cn), exergy efficiency, exergy losses and other properties were calculated for each case by changing the output pressure of the compressor of the critically important first-stage cycle, P11. COP values, compressor work (W C), heat energy dissipated in the condenser (Q cn), and exergy efficiency, exergy losses and other properties were calculated for each stage of the three-stage cycle, that is, for all three subcycles. Local optima are obtained and discussed here. Global optimums are not sought, and thermoeconomic analysis is not performed. Thermodynamic global optima, thermoeconomic analysis, and thermoeconomic optimization are not covered. Cryogenic cooling processes with ready-made liquids such as liquid nitrogen and liquid helium are also excluded. The subject of obtaining power from cold energy, which is a separate field, has not been addressed.
Since the COP value is the ratio of the heat absorbed to the given work and the heat absorbed is certain, the change in the COP value is examined by changing the amount of work given here. Calculations are made by reducing the throttling valve outlet pressure of the second subcycle at certain rates. At P 7 = 4,590 kPa, the throttling valve outlet pressure is 80 kPa, and the temperature is −88°C, and the pressure and temperature values are reduced at certain rates, and the material, phase, mass flow, temperature, pressure, enthalpy, entropy, energy and exergy of the flow lines are calculated from these values. The temperature was reduced to approximately 5°C below the condenser outlet temperature. As a result of the calculations, it has been observed that the most optimum data are the situation when the throttling valve outlet pressure is 30 kPa.
Since the given amount of work is used to compress the refrigerant through the compressor, the compression ratio was changed and the analysis was performed for different outlet pressures. Since the first subcycle is more effective than the second subcycle, the operating conditions of the first subcycle were changed by changing the compressor outlet pressure. Thermodynamic properties of flow lines for natural gas entering at 20°C and 400 kPa (P 7 = 2,400 kPa) are given in Table 6.
Thermodynamic properties of flow lines for natural gas entering at 20°C and 400 kPa (P 7 = 2,400 kPa)
| Flow Num. | Substance | Phase | Mass flow (kg/s) | Temp. (°C) | Pres. (kPa) | Enthalpy (kJ/kg) | Entropy (kJ/kg K) | Energy (kW) | Exergy (kW) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | Water | Liquid | 43.73 | 15 | 200 | 63 | 0.225 | −1,825 | 27.4 |
| 2 | Water | Liquid | 43.73 | 90 | 200 | 377 | 1.1928 | 11,904 | 1,135 |
| 3s | Ethane | Gas | 23.82 | 159 | 3,800 | 777 | 2.4835 | 2,629 | 9,127 |
| 3 | Ethane | Gas | 23.82 | 178 | 3,800 | 826 | 2.5948 | 3,785 | 9,492 |
| 4 | Ethane | Liquid | 23.82 | 20 | 3,800 | 201 | 0.733 | −11,099 | 7,834 |
| 5 | Ethane | Liquid | 23.82 | −108 | 30 | 201 | 1.404 | −11,099 | 5,195 |
| 6 | Ethane | Gas | 23.82 | −108 | 30 | 341 | 2.2549 | −7,761 | 360 |
| 7s | Methane | Gas | 4.62 | 10 | 2,400 | 853 | 4.8603 | −264 | 2,245 |
| 7 | Methane | Gas | 4.62 | 27 | 2,400 | 891 | 4.9901 | −88 | 2,242 |
| 8 | Methane | Liquid | 4.62 | −104 | 2,400 | 227 | 1.577 | −3,160 | 3,877 |
| 9 | Methane | Liquid | 4.62 | −162 | 80 | 227 | 2.086 | −3,160 | 3,173 |
| 10 | Methane | Gas | 4.62 | −162 | 80 | 511 | 4.6954 | −1,845 | 893 |
| 11 | N.Gas- Methane | Gas | 2 | 20 | 400 | 896 | 5.9177 | −28 | 427 |
| 12 | N.Gas- Methane | Gas | 2 | −82 | 400 | 674 | 4.9873 | −473 | 537 |
| 13 | N.Gas- Methane | Liquid | 2 | −157 | 400 | 16 | 0.1351 | −1787.744 | 2,116 |
Energy balance, COP exergy efficiency, and exergy losses for natural gas entering at 20°C and 400 kPa (P 7 = 2,400 kPa) are given in Table 7.
Energy balance, COP exergy efficiency, and exergy losses for natural gas entering at 20°C and 400 kPa (P 7 = 2,400 kPa)
| 1. Subcycle | COP1 = 0.833, W C1S = 1579.1 kW | Q Cn1 = 2894.1 kW, E Cn1los = 3088.6 kW | W C1 = 1754.6 kW, X 9 = 0.459, E TV1los = 704.3 kW |
| Energy balance Q Cn1 ≈ Q e1 + W C1S | |||
| 2894.1 ≈ 1315 + 1579.1 = 2894.1 | |||
| E C1los = 406.5 kW | |||
| E TV1los = 701.4 kW | |||
| E TOT1los = 4900.8 kW | |||
| 2. Subcycle | COP2 = 0.321, W C2S = 10390.3 kW | Q C2 = 13728.8 kW, E Cn2los = 551.1 kW | W C2 = 11544.8 kW, X 5 = 0.727, E TV2los = 2638.4 kW |
| Energy balance Q Cn2 ≈ Q e2 + W C2S | |||
| 13728.8 ≈ 3338.5 + 10390.3 = 13728.8 | |||
| E C2los = 2,413 kW | |||
| E TOT2los = 5602.5 kW | |||
| Cycle | COPT = 0.279 W CTOTS = 11969.4 kW | Q Cn2 = 13728.8 kW, E Cnev1-2Tlos = 3639.7 kW, η Tcycle, Ex = 0.235 | W CTOP = 13,299 kW, E TVTlos = 3342.7 kW |
| Energy balance Q CnT ≈ Q eT + W CTOTS | |||
| 13728.8 ≈ 11969.4 + 1759.4 = 13728.8 | |||
| E CTlos = 2819.5 kW | |||
| E TOTlos = 10503.3 kW |
W C1, W C2, powers, and total compressor W Call power curves of the whole cycle obtained for different output pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate are given in Figure 3.
W C1, W C2, powers, and total compressor W kAll power curves of the whole cycle obtained for different output pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate.
As shown in Figure 3, operating the first-stage compressor at high pressures increases the amount of work consumed in the subcycles and therefore the entire cycle. Therefore, in order to reduce the work consumption and increase the efficiency of vapor compression cascade cycles, low pressures should be operated and the work obtained by using a suitable turbine instead of a throttling valve should be reduced with the compressor work. It is also seen that operating the second-stage compressor at around 2,400 kPa is the optimum operating pressure.
The COP1, COP2, and COPall curves of the whole cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate are given in Figure 4. As the pressure of the first-stage compressor increases, the COP value decreases rapidly, but as the pressure of the second-stage compressor increases, the COP value increases. The COP value of the whole cycle increases as the pressure of the first-stage compressor increases. This increase ranges from 0.2 to 0.35 (75% increase).
COP1, COP2, and COPall curves of the whole cycle obtained for different outlet pressures at which the first- stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate.
In Figure 5, the exergy efficiency curve of the entire cycle is obtained for different outlet pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate. It is seen that the exergy efficiency is maximum at values between 2,400 and 2,800 kPa at the outlet pressure of the first-stage compressor and decreases at other pressures. The lowest exergy efficiency occurs at pressures close to the critical pressure.
Exergy efficiency curves of the whole cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate.
In Figure 6, the exergy losses curve of the entire cycle is obtained for different outlet pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate. It is seen that the operating pressure of the first-stage compressor with the least exergy losses is between 2,000 and 2,800 kPa.
Exergy loss curves of the entire cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression two-stage cascade cryogenic cycle can operate.
When Figures 3–6 are interpreted together, it can be seen from Figure 3 that the minimum total work required for cryogenic cooling is around P 7 = 2,400 kPa. Here, the second lower stage pressure is taken as P 5 = 30 kPa. Although it is seen from Figure 4 that the best COP value is 0.35 near the critical pressure, the COP value is 0.30 between P 7 = 2,400 and 2,800 kPa. Figure 5 shows that the maximum exergy efficiency was obtained around P 7 = 2,400 kPa (0.235). Figure 6 shows that the minimum exergy loss is obtained between P 7 = 2,000 and 2,800 kPa (around 10,500 kW). Considering that the work given to the compressor is valuable and the heat energy taken from the condenser is less valuable, it is seen that it is logical and profitable to work close to the minimum loss, maximum exergy efficiency, and maximum COP value.
3.2 Three-stage vapor compression cascade cryogenic cycle
Here, the change in the COP value was examined by changing the amount of work given. Since the given amount of work is used to compress the refrigerant through the compressor, the compression ratio was changed and the analysis was performed for different outlet pressures. Since the first subcycle is more effective than the second and third subcycles, the operating conditions of the first subcycle are examined by changing the compressor outlet pressure.
In Table 8, the thermodynamic properties of the flow lines (P 11 = 1,200 kPa) are given for the natural gas entering at 20°C and 400 kPa pressure. Here, the outlet pressure of the first subcycle is reduced by 400 kPa compared to the previous table, and the outlet pressure is taken as 1,200 kPa. In Table 9, the energy balance, COP exergy efficiency, and exergy losses (P 11 = 1,200 kPa) are calculated from the values in Table 8 for the case of natural gas entering at 20°C and 400 kPa pressure.
Thermodynamic properties of flow lines for natural gas entering at 20°C and 400 kPa (P 11 = 1,200 kPa)
| Flow Num. | Substance | Phase | Mass flow (kg/s) | Temp. (°C) | Pres. (kPa) | Enthalpy (kJ/kg) | Entropy (kJ/kg K) | Energy (kW) | Exergy (kW) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | Water | Liquid | 29.24 | 15 | 200 | 63 | 0.2246 | −1,219 | 27 |
| 2 | Water | Liquid | 29.24 | 105 | 200 | 440 | 1.3634 | 9,809 | 1,125 |
| 3s | Propane | Gas | 22.00 | 110 | 2,000 | 649 | 2.0565 | 399 | 5,588 |
| 3 | Propane | Gas | 22.00 | 120 | 2,000 | 674 | 2.1187 | 952 | 5,733 |
| 4 | Propane | Liquid | 22.00 | 20 | 2,000 | 147 | 0.5473 | −10,631 | 4,458 |
| 5 | Propane | Liquid | 22.00 | −42 | 80 | 147 | 0.654 | −10,631 | 3,758 |
| 6 | Propane | Gas | 22.00 | −42 | 80 | 422 | 1.8716 | −4,577 | 1,824 |
| 7s | Ethane | Gas | 7.72 | 157 | 4,800 | 762 | 2.3887 | 732 | 3,056 |
| 7 | Ethane | Gas | 7.72 | 176 | 4,800 | 811 | 2.5033 | 1,113 | 3,173 |
| 8 | Ethane | Liquid | 7.72 | −37 | 4,800 | 10 | 0.0073 | −5,072 | 2,735 |
| 9 | Ethane | Liquid | 7.72 | −127 | 8 | 10 | 0.419 | −5,072 | 1,787 |
| 10 | Ethane | Gas | 7.72 | −125 | 8 | 321 | 2.4852 | −2,343 | −573 |
| 11s | Methane | Gas | 3.62 | −38 | 1,200 | 756 | 4.829 | −556 | 1,443 |
| 11 | Methane | Gas | 3.62 | −26 | 1,200 | 784 | 4.9414 | −458 | 1,420 |
| 12 | Methane | Liquid | 3.62 | −122 | 1,200 | 147 | 1.1045 | −2,757 | 3,258 |
| 13 | Methane | Liquid | 3.62 | −162 | 80 | 147 | 1.3573 | −2,757 | 2,986 |
| 14 | Methane | Gas | 3.62 | −162 | 80 | 511 | 4.6954 | −1,442 | 698 |
| 15 | N.Gas – Methane | Gas | 2 | 20 | 400 | 896 | 5.9177 | −28 | 427 |
| 16 | N.Gas – Methane | Gas | 2 | −37 | 400 | 771 | 5.4437 | −278 | 459 |
| 17 | N.Gas – Methane | Liquid | 2 | −82 | 400 | 674 | 4.9873 | −473 | 537 |
| 18 | N.Gas – Methane | Liquid | 2 | −157 | 400 | 16 | 0.1351 | −1,788 | 2,116 |
Energy balance, COP exergy efficiency, and exergy losses for natural gas entering at 20°C and 400 kPa (P 11 = 1,200 kPa)
| 1. Subcycle | COP1 = 1.48 | W C1S = 887.7 kW, Q Cn1 ≈ Q e1 + W C1S, E Cn1ev2los = 443.6 kW | Q Cn1 = 2202.7 kW, E TV1los = 272.8 kW | X 13 = 0.305, E EV1los = 708.7 kW |
| W C1 = 986.3 kW | ||||
| Energy Balance | ||||
| 2202.7 ≈ 1315.024 + 887.7 = 2202.7 | ||||
| E C1los = 264.5 kW | ||||
| E TOT1los = 1689.6 kW | ||||
| 2. Subcycle | COP2 = 0.704 | W C2S = 3406.6 kW, Q Cn2 ≈ Q e2 + W C2S | Q Cn2 = 5803.6 kW, X 9 = 0.419, E Cn2ev3los = 2339.4 kW | W C2 = 3785.1 kW, E TV2los = 948.1 kW |
| Energy balance | ||||
| 5803.6 ≈ 2,397 + 3406.6 = 5803.6 | ||||
| E C2los = 39.1 kW | ||||
| E TOT2los = 3326.6 kW | ||||
| 3. Subcycle | COP3 = 1.216 | W C3S = 4975.7 kW, Q Cn3 ≈ Q B3 + W C3S | Q Cn3 = 11029 kW, W C3 = 5528.6 kW, E Cn3los = 176.2 kW | Q Ngas = 17591.4 kW, X 5 = 0.380, E TV3los = 700.9 kW |
| Energy balance | ||||
| 11,029 ≈ 6053.7 + 4975.7 = 11,029 | ||||
| E C3kay = 1619.9 kW | ||||
| E TOT3los = 2,497 kW | ||||
| Cycle | COPT = 0.653 | W CTOTS = 9,270 kW, Q Cnt ≈ Q et + W CTOTS, E CnTlos = 2959.2 kW, η Tcycle, Ex = 0.300 | Q Cn3 = 11,029 kW, E TVTlos = 1921.8 kW | E eTlos = 708.7 kW |
| W CTOT = 10,300 kW | ||||
| Energy balance | ||||
| 11,029 ≈ 9,270 + 1759.14 = 11,029 | ||||
| E CTlos = 1923.5 kW | ||||
| E TOTlos = 7513.2 kW |
As shown in Figure 7, operating the first-stage compressor at high pressures decreases the COP value of the first cycle, but increases the COP values of the second cycle rapidly. It slightly increases the COP values of the third cycle. Considering the COP values of the entire cycle, operating the first-stage compressor at high pressure decreases the COP values of the entire cycle. If it is remembered that the COP value is the ratio of the carried heat to the given compressor work, it is clear that operating the compressor with less power at low pressures will increase the COP values. It has been observed that this logical result is also obtained in the curves.
COP1, COP2, COP3, and COPall curves of the whole cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
The total compressor loss curve obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate is given in Figure 8. As shown in the figure, operating the first-stage compressor at high pressures increases the total losses of the entire cycle from 7,500 to 18,550 kW. The increase in total exergy losses is around 247%.
Total compressor losses curve obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
The exergy efficiency curve of the entire cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate is given in Figure 9. As shown in the figure, operating the first-stage compressor at high pressures increases the exergy efficiency of the entire cycle. The increase in total exergy efficiency is around 160%.
Exergy efficiency curve of the entire cycle obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
Looking at the curves and results in Figures 7–9, it becomes clear that high pressures should not be operated. Working at low pressures as much as possible increases efficiency as less work is required.
Likewise, when the relevant tables and figures are examined, serious work loss occurs during expansion to low pressure in the throttling valve. Since this situation will cause great losses in large cycles, a suitable (expansion) power turbine is placed instead of the throttling valve, so that work is obtained during expansion to low pressure. The work obtained from the turbine is spent on the compressor, thus increasing the efficiency by reducing the work to be given to the compressor from outside. Many large-scale natural gas liquefaction cycles replace the throttling valve with a suitable turbine and reduce expansion losses. Such an expansion turbine is available in the Claude cycle, which is an efficient cycle.
As shown in Figure 11, the total exergy losses of the entire cycle are the point where the second-stage compressor is operated at P 7 = 1,200 kPa. Exergy losses increase by 40% from the point where it is operated at P 7 = 1,200 kPa to the point where it is operated at P 7 = 4,400 kPa. Therefore, it is necessary to operate the second-stage compressor at as low pressure as possible for low exergy losses.
As shown in Figure 12, the maximum exergy efficiency of the entire cycle is the point at which the second-stage compressor is operated at P 7 = 1,200 kPa. The exergy efficiency decreases by 18.5% when it is operated at P 7 = 1,200 kPa from the point where it is operated at P 7 = 4,400 kPa. Therefore, it is necessary to operate the second-stage compressor at as low pressure as possible for maximum exergy efficiency.
3.3 Local optimization of two- and three-stage vapor compression cascade cryogenic cycles results and discussion
In the two-stage cycle, when Figures 3–6 are examined and the relevant tables are examined, it is seen that the local optimum values emerge spontaneously. Here, the optimum points for us are the points where the minimum work, minimum exergy loss and maximum COP values and maximum exergy efficiency. The points where the two minimum and two maximum points intersect are the optimum points we are looking for. The cycle also needs to be run at these optimum points. It can be seen from Figure 3 that the minimum total work required for cryogenic cooling is around P 7 = 2,400 kPa. Here, the second lower stage pressure is taken as P 5 = 30 kPa. Although it is seen from Figure 4 that the best COP value is 0.35 near the critical pressure, it is seen that the COP value is 0.30 between P 7 = 2,400 and 2,800 kPa. It can be seen from Figure 5 that the maximum exergy efficiency was obtained around P 7 = 2,400 kPa (0.235). It is seen from Figure 6 that the minimum exergy loss is obtained between P 7 = 2,000 and 2,800 kPa (around 10,500 kW). The two minimum compressor work and the minimum exergy loss coincide with the maximum exergy loss, which is one of the two maximums, at the same point, that is, at the point where P 7 = 2,400 kPa. The point where the COP is maximum is 0.35 at the critical pressure P 7 = 4,590 kPa, and the COP value is 0.3 around P 7 = 2,400 kPa, where the other three optimum points are obtained. Considering that the work given to the compressor is valuable and the heat energy taken from the condenser is less valuable, it is seen that it is reasonable and profitable to work close to the minimum loss, minimum compressor work, maximum exergy efficiency, and maximum COP value.
The points where the two minimum and two maximum points intersect are the optimum points we are looking for. The cycle also needs to be run at these optimum points. It can be seen from Figure 8 that the least compressor losses are obtained at P 11 = 1,200 kPa. It can be seen from Figure 7 that the maximum COP value is obtained at P 11 = 1,200 kPa. From Figure 8, the minimum exergy losses of the whole cycle are obtained at P 11 = 1,200 kPa. From Figure 9, the maximum exergy efficiency of the whole cycle is again obtained at P 11 = 1,200 kPa. The intersection of the two minimum and two maximum points is obtained at P 11 = 1,200 kPa. Here, our optimum point shows us that it should be operated at as low pressures as possible.
Figures 10–12 show how the operation of the second subcycle or the second sub-stage in the three-stage cycle, that is, the different pressures of the second subcycle compressor (taking the first-stage compressor pressure constant) affect the whole cycle. When the outlet pressure of the second-stage compressor P 7 is changed between 1,200 and 4,400 kPa, the COP value of the whole cycle increases by around 2% at P 7 = 1,200 kPa, as shown in Figure 10.
COP1, COP2, COP3, and COPall curves of the whole cycle obtained for different outlet pressures at which the second-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
Exergy loss curve of the entire cycle obtained for different outlet pressures at which the second-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
Exergy efficiency curve of the entire cycle obtained for different outlet pressures at which the second-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate.
4 Conclusions
In this study, optimum operating conditions are obtained by applying exergy analysis and local optimization methods to some cycles of the cryogenic field. The first and second laws of thermodynamics, exergy analysis, and local optimization methods were applied to the two- and three-stage vapor compression cascade cryogenic cycles, which are the most widely used methods. In this study, the mass flow rate, pressure, temperature, enthalpy, entropy, energy, and exergy values of each device and cycle flow points in the two- and three-stage vapor compression cascade cryogenic cycle cycles are found, and energy balances are established for each operating condition and given in tables. The obtained findings and results revealed the optimum working conditions of these cycles. It has been understood that in order to obtain high efficiency and high COP values, the compressor compression ratio should be reduced as much as the working fluid allows and a turbine should be used instead of the throttling valve in order to convert the work losses in the throttling valve into gain. The high compression ratio of the compressor increases the work spent on the cycle and the amount of heat dissipated, and decreases the COP values of the cycle. The work obtained by using the turbine instead of the throttling valve is spent on the compressor work, thus reducing the amount of work required by the cycle from the outside. For this reason, conversion efficiency and COP values can be increased.
If the two-stage cycle is interpreted together with Figures 3–6, it can be seen from Figure 3 that the minimum total work required for cryogenic cooling is around P 7 = 2,400 kPa. Here, the second lower stage pressure is taken as P 5 = 30 kPa. Although it is seen from Figure 4 that the best COP value is 0.35 near the critical pressure, it is seen that the COP value is 0.30 between P 7 = 2,400 and 2,800 kPa. Figure 5 shows that the maximum exergy efficiency was obtained around P 7 = 2,400 kPa (0.235). It is seen from Figure 6 that the minimum exergy loss is obtained between P 7 = 2,000 and 2,800 kPa (around 10,500 kW). Considering that the work given to the compressor is valuable and the heat energy taken from the condenser is less valuable, it is seen that it is logical and profitable to work close to the minimum loss, maximum exergy efficiency, and maximum COP value.
As shown in Figure 7, operating the first-stage compressor at high pressures decreases the COP value of the first cycle, but increases the COP values of the second cycle rapidly. It slightly increases the COP values of the third cycle. The total compressor loss curve obtained for different outlet pressures at which the first-stage compressor of the vapor compression three-stage cascade cryogenic cycle can operate is given in Figure 8. As shown in the figure, operating the first-stage compressor at high pressures increases the total losses of the entire cycle from 7,500 to 18,550 kW. The increase in total exergy losses is around 247%. As shown in Figure 9, operating the first-stage compressor at high pressures increases the exergy efficiency of the entire cycle. The increase in total exergy efficiency is around 160%.
In the three-stage cycle, the operation of the second sub-stage, that is, the high pressures of the second subcycle compressor (taking the first-stage compressor pressure constant) adversely affects the performance of the entire cycle. When the second-stage compressor is operated at low pressure, the COP value increases by 2%, the exergy efficiency increases by 20%, and the exergy losses decrease by around 40%.
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Funding information: Authors state no funding involved.
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Author contributions: The authors have equal rights on the preparation of this article.
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Conflict of interest: Authors state no conflict of interest.
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Ethical approval: The conducted research is not related to either human or animals use.
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Data availability statement: The processed data necessary to reproduce these findings are available upon request with permission.
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