## Abstract

Engineers and scientists are continuously in search of higher power system efficiencies. Among new ones, supercritical recompression carbon dioxide power cycle has been promising. In addition to the simple recompression cycle, modified versions of supercritical recompression carbo have been introduced. These modified versions are Recompression Reheating cycle, Recompression Partial Cooling cycle, Recompression Partial Cooling with Reheating cycle, Recompression Intercooling cycle, and Recompression Intercooling with Reheating cycle. This paper investigates performances of the modified recompression cycles by developing an extensive thermodynamic model for this purpose. For these analyses, many parameters such as isentropic efficiencies of compressors and turbines, effectiveness of energy exchangers, maximum and minimum pressures, and temperatures within the cycle have been kept constant. It is also assumed that the temperature of the source of energy is 600 °C. This temperature selection is based on the operational temperatures typical of current solar thermal, nuclear, and biomass/waste energy generation technologies. Parametric studies using intermediate pressure and split ratio have been done to determine the optimum values resulting in the maximum efficiencies of these cycles. The solution of the thermodynamic model requires solving simultaneous energy, entropy, and exergy balance equations. The results show three cycles have very close maximum efficiency. These are Recompression Reheating cycle, Recompression Intercooling with Reheating cycle, and Recompression Intercooling cycle having thermal efficiencies of 39.61%, 39.57%, and 39.49%, respectively. The Recompression Intercooling with Reheating cycle has the highest net-work among the above cycles when operating at their maximum thermal efficiencies.

## 1 Introduction

Supercritical carbon dioxide offers promising advantages over traditional fluids like water and air as working fluid for power cycles. These cycles, especially in their supercritical phase, have shown potential for enhanced efficiency and more compact designs in experimental settings [1–3]. Among various configurations, the Recompression sCO_{2} cycle, along with its variants, is a central area of investigation [4–8]. These cycles are known for their high efficiency, even with lower turbine inlet temperatures and compressor pressure ratios. However, they offer a compromise in terms of net-work output when compared to simpler Brayton cycles [9]. Research efforts are directed towards modifications such as intercooling [10–13], partial cooling [14–16], reheating [17,18], and their combinations [19] to optimize these cycles for different energy sources [20–23] and achieve a balance between efficiency and power output. Considering these modifications, particularly the introduction of intermediate compression or expansion stages, several research studies have focused on the optimization of intermediate pressure. These studies aim to refine the pressure adjustments in various recompression cycles in order to enhance their efficiency and operational flexibility under different conditions [11,24,25]. Additionally, current studies delve into the exergy destruction in all components within the power plant [26–28]. Few studies have been concentrating on the design of recuperators and precoolers. However, the cost of such designs, particularly the energy exchangers, remains a significant challenge [29–31]. Recognizing the complexity of intermediate pressure and its impact, this research advances beyond previous simplifications that correlated intermediate pressure with high and low pressures within the cycle. Instead, this paper explores the combined effect of intermediate pressure and split ratio on cycle optimization and performance.

Section 2 outlines the thermodynamics model including reheating, partial cooling, and intercooling along with their integrated forms. The equations used to calculate temperatures at each state are described. The total number of nonlinear equations to be solved ranges from 11 to 15 depending on the design of the power plant. The Newton–Raphson method, executed in python, is employed to solve these equations, with fluid properties sourced from the open-access CoolProp library [32,33].

Section 3 outlines individual cycle configurations and determines specific net-work output, thermal, and exergetic efficiency as a function of intermediate pressure around best-split ratios. It then presents a comparative study on maximum efficiency and output as a function of optimum intermediate pressure and split ratio. Additionally, it further compares exergy destruction across components when the cycle operates at the highest efficiency.

Section 4 describes the conclusion of this research.

## 2 Thermodynamics Model

_{2}cycle configurations. For a given state k, characterized by temperature $Tk$ and pressure $pk$, the enthalpy $hk$ and entropy $sk$ are determined by:

Simple Recompression Cycle (*R*) is shown in Fig. 1. In this cycle, the working fluid goes through processes such as precooled main compression, recompression, heating-expansion, recuperation, mixing, and splitting. This setup is the reference model for all other designs. To keep things uniform, the same numbering system is used in all cycles for main parts such as compressor, turbine, recuperators, and mixing chamber.

Subsequent configurations, such as Reheating (RRH), Partial Cooling (RPC), Intercooling (RI), and their combinations such as Partial Cooling with Reheating (RPCR) and Intercooling with Reheating (RIR), are modified designs of Recompression cycle. Each variant introduces additional states, pressure variations, and components such as a boost compressor (BC) with additional precooling, and a low-pressure turbine (LPT) with additional reheating.

*Reheating Cycle (RRH)*: This configuration improves the recompression cycle by adding an extra energy exchanger that interacts with a constant temperature energy source as shown in Fig. 2. This represents the additional energy input into the cycle that takes place at intermediate pressure before the low-pressure turbine.

*Partial Cooling Cycle (RPC)*: Intermediate pressure in this cycle affects both re-compressor (RC) and MC as the exit of the BC feeds into both, as shown in Fig. 3.

*Partial Cooling with Reheating Cycle (RPCR)*: This design merges the characteristics of RPC and RRH and is shown in Fig. 4.

*Intercooling Cycle (RI)*: RI, shown in Fig. 5, also includes a BC. The difference lies in the location of the intermediate pressure, which in this cycle occurs only at the inlet of the main compressor connected to an intercooler.

*Intercooling with Reheating Cycle (RIR)*: This design merges the characteristics of RI and RRH and is shown in Fig. 6.

Additional thermodynamics relations around the cycles are as follows:

Temperatures within the system are constrained such that $Tmin<T<Tmax,$ where $Tmin$ is the MC and BC inlet temperature and $Tmax$ is the temperature at the inlet of the HPT and LPT. Additionally, to ensure the cycle's validity, the cold side outlet temperature of the LTR, $Ti,cold$, must be kept below its hot inlet temperature, $Ti,hot$. The number of unknown temperatures depends on the cycle. All of these cycles are recompression, and they have different sets of unknown temperatures. For a base Recompression cycle (R), the unknowns are $T=[T2,T2s,T3,T4,T5,T7,T7s,T8,T9,T12,$$T12s]$.

For enhanced configurations such as RI and RPC, additional temperature states are introduced to account for the effects of intercooling or partial cooling which leads to the inclusion of $T14$ and $T14s$ in the unknown temperature list: $T=[T2,T2s,T3,T4,T5,T7,T7s,$$T8,T9,T12,T12s,T14,T14s]$. Moreover, in configurations involving reheating, such as RIR and RPCR, further complexities are added with the introduction of $T16$ and $T16s$ to the unknowns which extends the list to $T=[T2,T2s,T3,T4,T5,T7,T7s,T8,T9,$$T12,T12s,T14,T14s,T16,T16s]$. For Reheating (RRH) only, the addition is the temperatures at state 16, which results in an unknown temperature list of $T=[T2,T2s,T3,T4,T5,T7,T7s,$$T8,T9,T12,T12s,T16,T16s]$. To determine these temperatures, 11 to 15 nonlinear equations have been solved. These equations focus on the changes in enthalpy, dictated by the thermodynamic equations, as a method to model and solve for temperatures. These unknown temperatures are calculated using the Newton–Raphson method, aided by the CoolProp library for thermodynamic property evaluations.

## 3 Results and Discussions

Results from cycle configurations are presented and analyzed in this section. The analysis is conducted under a set of constant parameters. Values of parameters are shown in Table 1.

Parameter | Value |
---|---|

Compressors isentropic efficiency ($\eta comp$) | 0.85 |

Turbine isentropic efficiency ($\eta turb$) | 0.85 |

Effectiveness of low-temperature recuperator ($\epsilon LTR$) | 0.85 |

Effectiveness of high-temperature recuperator ($\epsilon HTR$) | 0.85 |

Precooler outlet temperature ($Tmin$) | 32 °C |

Turbine inlet temperature ($Tmax$) | 550 °C |

Low pressure ($plow$) | 75 bar |

High pressure ($phigh$) | 200 bar |

Intermediate pressure ($pint$) | 76 to 180 bar |

Split ratio ($rs$) | 0.3 to 1 |

Source temperature ($Ts$) | 600 °C |

Environment temperature ($To$) | 27 °C |

Parameter | Value |
---|---|

Compressors isentropic efficiency ($\eta comp$) | 0.85 |

Turbine isentropic efficiency ($\eta turb$) | 0.85 |

Effectiveness of low-temperature recuperator ($\epsilon LTR$) | 0.85 |

Effectiveness of high-temperature recuperator ($\epsilon HTR$) | 0.85 |

Precooler outlet temperature ($Tmin$) | 32 °C |

Turbine inlet temperature ($Tmax$) | 550 °C |

Low pressure ($plow$) | 75 bar |

High pressure ($phigh$) | 200 bar |

Intermediate pressure ($pint$) | 76 to 180 bar |

Split ratio ($rs$) | 0.3 to 1 |

Source temperature ($Ts$) | 600 °C |

Environment temperature ($To$) | 27 °C |

*Specific Net-work Output Analysis for Individual Configurations*: Specific net-work output as a function of intermediate pressure for different split ratios have been calculated and presented in this section. It has been observed that for a fixed intermediate pressure, greater split ratios lead to higher net-work output due reduced recompression stage. All cycles exhibit an optimum intermediate pressure beyond which maximum net-work output begins to decline.

Figure 7 shows specific net-work output as a function of intermediate pressure ($p15$) for Reheating cycle (RRH). By increasing the number of reheating the working fluid, the overall expansion process approaches isothermal expansion more closely, which generates more work than adiabatic expansion. For an ideal gas, when the pressure ratio of both turbines is equal, the output is maximized which corresponds to the optimum intermediate pressure in this cycle. It can be seen that the maximum net-work happens when the split ratio is 0.73 and the intermediate pressure is 123 bar.

Figure 8 shows the specific net-work output as a function of intermediate pressure ($p14$) in RPC. The net-work is maximum at an intermediate pressure of approximately 83 bar, regardless of the split ratio. The net-work output declines as the intermediate pressure either increases or decreases.

Figure 9 shows specific net-work output as a function of intermediate pressures ($p14=p15$) in a Partial Cooling and Reheating (RPCR) cycle. This design attains the greatest output in comparison to other cycles. The addition of reheating to the RPC increases the work output by reintroducing energy at a higher temperature into the cycle. This added stage of energy input shifts the optimum intermediate pressure closer to that of RRH. Maximum specific net-work happens when the intermediate pressure is 108 bar and the split ratio is 0.69.

Figure 10 shows specific net-work output against intermediate pressures ($p14$) in RI. The same trend is observed in RPC before optimum intermediate pressure. This indicates that effective cooling at these pressures benefits both cycles. Specifically, for RI, targeted cooling of the split flow leads to less energy being rejected when compared to RPC.

Figure 11 shows the relationship between specific net-work output and the intermediate pressures ($p14=p15$) in RIR. Due to an additional expansion stage, the net-work output is improved over RI. The peak occurs around an intermediate pressure of approximately 105 bar for the split ratio of 0.68.

Table 2 shows the best configurations for achieving maximum specific net-work output. RPCR leads with the highest specific net-work output of 103.44 kJ/kg at an intermediate pressure of 104.5 bar and a split ratio of 0.99. It is closely followed by RIR, which produces a specific net-work output of 103.17 kJ/kg. These configurations show a substantial increase in work output when combined with a high split ratio of 0.99, indicating the effectiveness of higher split ratios on the work output of the cycle. RPC, RI, and RRH configurations also show improved work outputs of 99.72, 99.34 kJ/kg and 96.74 kJ/kg, respectively, at the same split ratio, while at varying intermediate pressures, 83.4 bar and 83.6 and 123.3 bar, respectively. The Recompression cycle (R), while it does not require an intermediate pressure, falls behind with a specific net-work output of 91.31 kJ/kg.

Type of plant | Best intermediate pressure (bar) | Best-split ratio | Best of value of specific net-work output (kJ/kg) | Efficiency (%) |
---|---|---|---|---|

Recompression Partial Cooling with Reheating (RPCR) | 104.5 | 0.99 | 103.44 | 35.13 |

Recompression Intercooling with Reheating (RIR) | 104.9 | 0.99 | 103.17 | 35.17 |

Recompression Partial Cooling (RPC) | 83.4 | 0.99 | 99.72 | 35.20 |

Recompression Intercooling (RI) | 83.6 | 0.99 | 99.34 | 35.24 |

Recompression Reheating (RRH) | 123.3 | 0.99 | 96.74 | 37.25 |

Simple Recuperated | – | – | 91.75 | 30.60 |

Recompression (R) | – | 0.99 | 91.31 | 36.26 |

Type of plant | Best intermediate pressure (bar) | Best-split ratio | Best of value of specific net-work output (kJ/kg) | Efficiency (%) |
---|---|---|---|---|

Recompression Partial Cooling with Reheating (RPCR) | 104.5 | 0.99 | 103.44 | 35.13 |

Recompression Intercooling with Reheating (RIR) | 104.9 | 0.99 | 103.17 | 35.17 |

Recompression Partial Cooling (RPC) | 83.4 | 0.99 | 99.72 | 35.20 |

Recompression Intercooling (RI) | 83.6 | 0.99 | 99.34 | 35.24 |

Recompression Reheating (RRH) | 123.3 | 0.99 | 96.74 | 37.25 |

Simple Recuperated | – | – | 91.75 | 30.60 |

Recompression (R) | – | 0.99 | 91.31 | 36.26 |

### Thermal Efficiency and Exergetic Efficiency Analysis for Individual Configurations.

Thermal and exergetic efficiency have been calculated for all cycles having different values of intermediate pressure and split ratio. Results are shown as a function of intermediate pressure for a range of split ratios. It has been observed that for each cycle, the value of optimum intermediate pressure varies with the split ratio. For some values of split ratio resulting efficiencies are not smooth curves, and it could be due to thermodynamics properties of sCO_{2}. This will be discussed in detail in the following section. Thermal and exergetic efficiency of a Reheating (RRH) cycle at different intermediate pressures $(p15)$ is shown in Fig. 12. Values of split ratios are very close to the optimum value of 0.71. It can be seen that the maximum thermal efficiency will be about 40% while the maximum exergetic efficiency is over 60%.

In order to understand why there is a sharp edge for efficiency as a function of intermediate pressure for a split ratio of 0.71 as shown in Fig. 12, analysis has been made using air as a working fluid with the same input parameters in the Reheating cycle (RRH). The results are shown in Fig. 13. As can be seen, the thermal and exergetic efficiency of having air as a working fluid exhibits a smooth curve. This may mean that the thermodynamics properties of sCO_{2} may not be 100% correct. It also can be seen that using sCO_{2} in the power plant provides much better efficiency than air. This is true for all the configurations.

Figure 14 shows thermal and exergetic efficiency of a RPC at different intermediate pressures $(p15)$, around the optimum split ratio of 0.63. It can be seen that the maximum thermal efficiency of the cycle is around 37% while the maximum exergetic efficiency is 56%. The lower exergetic efficiency of RPC, compared to the Reheating cycle (RRH), is primarily due to the absence of a reheating process. Instead, RPC benefits from an enhanced energy rejection process, which effectively reduces the cycle's overall losses. The results show that there are sharp edges for performance characteristics of this cycle for some values of split ratio. Analogous to the analysis of Reheating cycle (RRH), the RPC performance has been evaluated using air as working fluid and results show that all performance characteristics follow smooth curves.

Figure 15 shows thermal and exergy efficiency of RPCR at different intermediate pressures $(p14=p15)$, around the optimum split ratio of 0.67. The graph indicates that the maximum thermal efficiency is approximately 37%, and the maximum exergy efficiency reaches about 57%. Analogous to the analysis of the Reheating cycle (RRH), RPCR performance has been evaluated using air as working fluid and results show that all performance characteristics follow smooth curves.

Figure 16 presents the impact of intermediate pressure $(p14)$ on the thermal and exergy efficiencies within an RI. The results show that the maximum thermal efficiency is approximately 39%, and the maximum exergy efficiency reaches about 60%. Enhanced cooling, as compared to the Partial Cooling (RPC) configuration, allows this cycle to achieve higher efficiency, thus enabling similar performance to cycle that employ reheating. Analogous to the analysis of the Reheating cycle (RRH), the RI performance has been evaluated using air as working fluid and results show that all performance characteristics follow smooth curves.

Figure 17 shows the thermal and exergy efficiency of the RIR at different intermediate pressures of $(p14=p15)$, focusing on the optimum split ratio of 0.69. This cycle exhibits the highest recorded thermal efficiency at approximately 39% and exergy efficiency at just over 60%. Analogous to the analysis of Reheating cycle (RRH) the RIR performance has been evaluated using air as working fluid and results show that all performance characteristics follow smooth curves.

Table 3 shows the optimum configurations for maximizing efficiency within different cycles. It reveals that Reheating cycle (RRH) achieves the best efficiency at 39.61% with an intermediate pressure of 144.3 bar and a split ratio of 0.71. This is closely followed by the RIR and RI with efficiencies of 39.57% and 39.49%, respectively. Notably, the Intercooling with RIR, with a lower intermediate pressure of 110.4 bar compared to Reheating (RRH), suggests a more balanced approach between pressure and splitting for enhanced efficiency. Recompression cycle (R) achieves a reasonable efficiency of 38.63% with a split ratio of 0.7. Partial Cooling configurations, both with Reheating (RPCR) and without RPC, show efficiencies of 37.97% and 37.35%, respectively, with intermediate pressures of 89.6 and 120.7 bar and split ratios of 0.67 and 0.63, respectively.

Type of plant | Best intermediate pressure (bar) | Best-split ratio | Best of value of efficiency (%) | Specific net-work output (kJ/kg) |
---|---|---|---|---|

RRH | 144.3 | 0.71 | 39.61 | 82.51 |

RIR | 110.4 | 0.69 | 39.57 | 92.20 |

RI | 78.4 | 0.67 | 39.49 | 84.72 |

R | – | 0.7 | 38.63 | 77.29 |

RPCR | 120.7 | 0.67 | 37.97 | 100.95 |

RPC | 89.6 | 0.63 | 37.35 | 96.59 |

Gas turbine with a single recuperator | – | – | 30.60 | 91.75 |

Type of plant | Best intermediate pressure (bar) | Best-split ratio | Best of value of efficiency (%) | Specific net-work output (kJ/kg) |
---|---|---|---|---|

RRH | 144.3 | 0.71 | 39.61 | 82.51 |

RIR | 110.4 | 0.69 | 39.57 | 92.20 |

RI | 78.4 | 0.67 | 39.49 | 84.72 |

R | – | 0.7 | 38.63 | 77.29 |

RPCR | 120.7 | 0.67 | 37.97 | 100.95 |

RPC | 89.6 | 0.63 | 37.35 | 96.59 |

Gas turbine with a single recuperator | – | – | 30.60 | 91.75 |

### Comparative Analysis.

In the comparative analysis, the points of maximum specific net-work output and maximum efficiency are used to determine the optimum split ratio and optimum intermediate pressure for each cycle. Figure 18 shows the relation between optimum intermediate pressure at each split ratio for maximum net-work output across different cycles. A color bar on the plots matches the optimum values of intermediate pressure at each split ratio. In Reheating cycle (RRH), as the split ratio increases optimum intermediate pressure remains the same to maintain the maximum net-work output. RPCR and the RIR cycles show a downward trend in the optimum intermediate pressure as the split ratio increases. For the RI, there is a noticeable decrease in the optimum intermediate pressure as the split ratio increases showing a sensitivity to the split ratio. Lastly, the RPC maintains a consistent optimum intermediate pressure across a range of split ratios, illustrating that its performance is less affected by how the flow is divided.

Figure 19 shows the maximum specific net-work output as a function of intermediate pressure at the maximum split ratio of 99% that yields the maximum net-work output independent of intermediate pressure. As the split ratio goes to 100%, recompression and in turn LTR do not operate, which makes RPCR and RIR to be the same as well as RI and RPC. Reheating cycle (RRH) proves to be useful only when combined with additional cooling-compression processes.

Figure 20 shows maximum thermal efficiency with respect to the split ratio at each intermediate pressure. A color bar on the plots matches the optimum values of split ratio at each intermediate pressure. For the Reheating cycle (RRH), the optimum split ratio increases from 0.71 to 0.74 as the intermediate pressure increases. RIR cycle's optimum split ratio slightly increases from 0.69 to 0.72 as the intermediate pressure increases. In the RI, the optimum split ratio varies to a greater extent, from 0.67 to 0.75 as the intermediate pressure increases. For the RPCR cycle, the plot shows an increase in the optimum split ratio, from 0.63 to 0.75, as intermediate pressure increases. The results for RPC show a maximum followed by a minimum and then increasing thermal efficiency, which is contrary to intuition. To learn more, these results have been investigated more. The performance of this cycle has been studied using air as the working fluid and results are shown in Fig. 21. This figure shows the thermal efficiency of this cycle using air and sCO_{2} as working fluids. As it can be seen the thermal efficiency for the cycle using supercritical carbon dioxide is higher than using air as a working fluid, which is true for all recompression cycles using sCO_{2} as a working fluid. Also, it can be seen that the results of thermal efficiency of this cycle using air as working fluid is smooth as a function of intermediate pressure, making this impression that there might be some errors in the property calculation of supercritical carbon dioxide.

Figure 22 shows the maximum thermal efficiency of different cycles as a function of split ratio with respect to intermediate pressure. A color bar on the plots matches the optimum values of intermediate pressure at each split ratio. It can be seen that the Reheating cycle (RRH) has the maximum thermal efficiency of 39.61% followed by RIR having a thermal efficiency of 39.57 and the RI with the efficiency of 39.49%. It can also be seen that the thermal efficiency of the Partial Colling cycle behaves abnormally. Again, this cycle like other ones has been analyzed using air as a working fluid and the results are shown in Fig. 23 showing smooth results for air as working fluid. Based on the air results someone would wonder if the properties of supercritical carbon dioxide are correct all the time.

## 4 Conclusion

This research investigates the effects of adjusting the intermediate pressure and split ratio on the performance and efficiency of sCO_{2}-modified recompression cycles. Additionally, the study gives insights into the relationship between intermediate pressure and split ratio at maximum efficiency and specific net-work output.

The Reheating cycle (RRH) stands out with the highest thermal efficiency with an optimum intermediate pressure of 144.3 bar, a split ratio of 0.71, a peak efficiency of 39.61%, and net-work of 82.51 kJ/kg. On the other hand, the RIR, with its optimum intermediate pressure set at 110.4 bar and a split ratio of 0.69, reaches a maximum efficiency close to that of the Reheating cycle (RRH) at 39.57% but distinguishes itself with the improvement in specific net-work output reaching 92.20 kJ/kg.

While the configurations with reheating and/or intercooling increase the performance of Recompression cycle (R) both in efficiency and the net-work output, they introduce complexity that translates to higher capital costs. The necessity of additional intercoolers, compressors, reheaters, and turbines for these configurations escalates the initial investment which requires an economic assessment to balance the efficiency gains against the increased expenditure.

The study shows that adding expansion and compression stages, especially close to the critical state, can maximize the specific net-work output. Nevertheless, this increases the cycle's sensitivity to pressure ratios and intermediate compression stages due to abrupt changes in the properties of sCO

_{2}. This sensitivity results in sudden and sharp peaks in efficiency observed across all cycles when specific combinations of intermediate pressure and split ratio are used. In contrast, cycles using air show a smooth transition in efficiency when adjusting split ratio and intermediate pressure This highlights the unique challenges of optimizing cycles that utilize sCO_{2}.Cycles with partial cooling applications including RPC and RPCR show a significant increase in net-work output at the expense of increased energy input by heat interaction which results in a reduction in overall efficiency when compared to Recompression cycle (R).

It is also found that the RPC reveals a complex relationship between optimum intermediate pressure and split ratio to maintain maximum efficiency. This is attributed to an error in CoolProp property data for a specific range of intermediate pressure and split ratio.

It is clear that increasing source temperature improves thermal and exergetic efficiencies. The decision to maintain a constant source temperature of 600 °C throughout the study is aligned with the typical operating temperature of current solar thermal, nuclear, biomass, and stored energy power plants.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.