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Article

The Dynamic Response Characteristics and Working Fluid Property Differences Analysis of CO2–Kr Mixture Power Cycle System

1
School of Energy and Power Engineering, Northeast Electric Power University, Jilin 132012, China
2
School of Mechanical Engineering, Chaohu University, Hefei 238024, China
*
Author to whom correspondence should be addressed.
Processes 2025, 13(6), 1735; https://doi.org/10.3390/pr13061735
Submission received: 8 April 2025 / Revised: 18 May 2025 / Accepted: 28 May 2025 / Published: 1 June 2025

Abstract

With the advancement of the energy transition, the thermodynamic degradation under high-load conditions and economic bottlenecks of the sCO2 Brayton cycle have become more prominent. CO2 mixture working fluids can improve system efficiency and economics through property optimization. However, the dynamic response characteristics of the system under disturbance factors are still unclear. Based on this, this paper establishes a dynamic model of the recompressed Brayton cycle for CO2 and CO2–Kr mixture. The dynamic behaviors of the two working fluids under mass flow, heat source power, and rotational speed disturbances are systematically compared, revealing the impact of the addition of Kr on the system’s dynamic response characteristics. From the perspective of the coupling mechanism in a mixture of working fluids, this paper further explores the reasons behind the differences in dynamic performance. The results show that mass disturbances have the most significant impact on the dynamic characteristics of the system. The response time of the turbine outlet temperature in the pure CO2 system is 15.43 s, with a temperature response amplitude of 12.32 K. When the system recovers to a steady state, the system’s efficiency and specific work are 30.37% and 42.52 kW/kg, respectively. In comparison, the CO2–Kr system demonstrates better dynamic performance, with the turbine outlet temperature response time reduced by 3.5 s and the temperature fluctuation amplitude decreased by 6.25 K. Additionally, the efficiency and specific work of the CO2–Kr system increased by 5.77% and 7.29 kW/kg, respectively. The introduction of Kr changes the physical property parameters of the working fluid, enhancing flow stability, and reducing pressure and temperature fluctuations, thereby improving the dynamic performance and disturbance resistance of the CO2–Kr system.

1. Introduction

With the advancement of the energy transition and the goal of carbon neutrality, the sCO2 Brayton cycle has become a research focus for advanced power cycle systems due to its high energy density, compact system structure, and environmentally friendly characteristics. However, this technology still faces multiple bottlenecks in large-scale applications. The thermodynamic performance of pure CO2 working fluid significantly degrades under high-load conditions, especially in high-temperature heat exchangers, where the abrupt change in the specific heat capacity of the fluid causes a sharp increase in exergy losses, leading to a decrease in system efficiency [1]. Furthermore, the system’s economic viability is constrained. The pure CO2 cycle relies on complex high-pressure equipment and precise temperature control technology to achieve high thermal efficiency, leading to high construction and operational costs [2]. These challenges have prompted researchers to explore new working fluid optimization solutions. Among these, the approach of adding additives to CO2-based working fluids has gradually become a key path to overcoming technological bottlenecks due to its unique property advantages [3].
In recent years, many researchers have studied the performance optimization of CO2-based mixture working fluid. Xia et al. [4] proposed two novel cooling and power systems with separators, combining a transcritical CO2 power cycle with an ejector refrigeration cycle. The system improved thermal performance and efficiency by using CO2-based mixtures as the working fluid. Xie et al. [5] established a recompressed sCO2 Brayton cycle system to recover exhaust heat from marine low-speed diesel engines. By optimizing five CO2-based mixture working fluids, they found that adding 20.11% propane to the mixture increased the net recovery by 17.02 kW, and adding 32.03% propane improved the cycle efficiency by 5.42%. Wang et al. [6] proposed a computer-aided molecular design method based on group contribution and perturbation chain statistical association fluid theory. Through molecular recognition models, they selected the optimal mixture to improve the performance of the CO2 transcritical power cycle. Mu et al. [7] established an improved CO2 transcritical Rankine cycle dynamic model, using CO2-based binary azeotropic mixtures as the working fluid. The results indicated that an appropriate pump speed could optimize the net output power and thermal efficiency. The CO2-based mixture system performed better at the optimal point, and adjusting the pump speed reduced the power of pure CO2 by 34 kW and 2.19%. Geng et al. [8] developed a new power cycle (NPC) that utilizes a CO2-based mixture of working fluid and downhole heat exchangers for the pressurized heat absorption process, improving cycle performance when the ground fluid temperature increased from 110 °C to 170 °C. The results showed that NPC outperformed the traditional ORC, especially at 130 °C, where the maximum net power output was higher. Yu et al. [9] optimized the thermodynamic performance of the Pressurized Heat Absorption Power Cycle (IPEPC) and Closed-Loop Geothermal Energy Extraction (CLGEE), finding that IPEPC outperformed other systems when the DHE length was less than 3 km, with the best performance at a DHE outer diameter of 0.22 m, guiding the geothermal industry. Niu and Ma et al. [10,11] were the first to use a thermodynamic analysis method based on the optimal split ratio to assess the potential application of CO2-based binary mixtures in solar power tower (SPT) systems. They analyzed the impact of key cycle parameters on the system’s thermal performance. The results showed that the CO2–Propane mixture improved thermal and exergy efficiency by 2.34% and 1.51%, respectively, compared to pure CO2. Xu et al. [12] conducted a multi-objective optimization evaluation of seven candidate organic fluids, and the results indicated that, with an increase in the mass fraction of the organic fluid, the performance improved significantly. The CO2/R32 and CO2/R161 systems outperformed other CO2-based mixtures. Wang et al. [13] studied the performance of three additives at different mass fractions and pure CO2, and the results showed that CO2 mixture fluids performed poorly in low-temperature hot water but significantly reduced condensation pressure. Compared to pure CO2, the mixture fluid increased the net output power by 2.45% and thermal efficiency by 19.46%, and it performed better at lower maximum pressures. Pan et al. [14] proposed a new transcritical power cycle using butane–CO2 and isobutane–CO2 as working fluids to address the CO2 condensation issue. The study showed that the optimal thermal efficiencies were 12.78% and 12.97%, and the combustible critical mole ratios for the two mixtures were 0.04/0.96 and 0.09/0.91, respectively.
It is important to note that power cycle systems inevitably face dynamic disturbances during operation. Load fluctuations, step changes in heat source power, and mass flow rate sudden changes caused by equipment start-ups and shutdowns can all disrupt the system’s thermal balance. Hao et al. [15] developed a dynamic model of a 1 MW gas–liquid type compressed CO2 energy storage system (GL-CCES). They analyzed the dynamic response during the liquid/gas tank discharge process. The results provided important insights for system design and safety control. Meng et al. [16] reported that propane-enriched CO2 mixtures achieved superior thermal efficiency, yet their operational stability was compromised under fluctuating conditions. Wang et al. [17,18,19] pioneered a hybrid energy architecture combining conventional coal-fired plants with supercritical CO2 power cycles. This hierarchical strategy achieved a 25% reduction in settling time compared to conventional single-loop controllers, with a 90% tracking accuracy for step-load perturbations. Meanwhile, Han and Zhou et al. [20,21] evaluated a solar-driven hybrid system incorporating solid oxide fuel cells and sCO2 technology, identifying significant seasonal performance disparities.
In addition, compared to other rare gases and hydrocarbons, Kr has some unique advantages. Kr has a lower critical temperature and critical pressure at low temperatures, making it more advantageous for low-temperature applications. Kr has moderate thermodynamic properties compared to Xe and can effectively operate in many low-temperature environments [22]. Kr is significantly less expensive than Xe and is more readily available, making it suitable for large-scale applications [23]. Furthermore, Kr has good chemical stability and high safety and does not exhibit the high reactivity or toxicity of certain hydrocarbons [24]. In summary, although Kr has clear advantages in terms of thermodynamic performance, cost, and stability, there is still a lack of in-depth understanding of the dynamic response characteristics of CO2-Kr mixture working fluid under dynamic conditions. The nonlinear characteristics of the mixture’s thermophysical parameters concerning component variations may exacerbate the risk of system dynamic instability, and the coupling mechanisms between different thermophysical parameters under disturbance conditions remain unclear.
Based on this, this study focuses on the recompressed Brayton cycle system, using CO2 and CO2–Kr mixture working fluid as the subjects. This study provides an in-depth analysis of the CO2–Kr mixture system from a dynamic perspective for the first time. By constructing a high-precision transient simulation model, the dynamic response characteristics of the CO2–Kr mixture and pure CO2 under mass flow, heat source power, and rotational speed disturbances were quantified, revealing the impact of Kr introduction on the system’s disturbance resistance. Compared to traditional steady-state analysis, this study not only reveals for the first time the differences between the mixture and pure CO2 in key operational parameters, specific work, and thermal efficiency but also explains, from the perspective of the coupling of physical properties, the interaction between properties and its effect on the system’s dynamic response characteristics. This research fills the gap in the dynamic response characteristics of CO2–Kr mixtures and provides important theoretical support for optimizing system dynamic characteristics and control system design. It contributes significantly to developing of thermodynamic conversion technologies towards high efficiency, stability, and low-cost practical applications with notable academic and practical value.

2. Mathematical Model

2.1. System Layout

The layout of the recompression cycle system is shown in Figure 1a, which includes the main compressor (MC), re-compressor (RC), turbine (Turb), high-temperature regenerator (HTR), low-temperature regenerator (LTR), and precooler (PC). Figure 1b presents the T-S diagram of the sCO2 cycle system. The entire cycle process is modeled and simulated using MATLAB Simulink, with the fluid thermophysical properties calculated using REFPROP 9.1 via MATLAB 2020b.

2.2. Thermodynamic Model and Assumptions

To simplify the system modeling and evaluate the dynamic response performance of CO2-based mixtures, the thermodynamic model is established based on the following assumptions:
(1)
The working fluid flow in all heat exchangers is one-dimensional.
(2)
No chemical reactions occur in the mixture.
(3)
The liquid resistance is neglected in pipes.
(4)
The cooling air used in the precooler is under ambient conditions.

2.2.1. Compressor and Turbine Model

Since various parameters such as density, thermal conductivity, and specific heat of CO2 change rapidly near the critical point, this imposes greater performance requirements on the compressor. Therefore, employing performance curves in modeling reflects the compressor’s performance characteristics more accurately [18]. The compressor performance curve is shown in Figure 2. The turbine model is calculated using Equations (1)–(9) [25], and the resulting performance curve is shown in Figure 3. In Figure 2 and Figure 3, m, N, P, and η represent the mass flow rate, rotational speed, pressure, and efficiency of the respective equipment, respectively. The subscript design denotes the design values of these parameters, while in and out indicate the inlet and outlet of the corresponding equipment. After normalization, the performance curves for the compressor and turbine are obtained.
The mass flow rate of the turbine model is
m t u r = ρ t u r , i n A n o z z l e C
C = 2 Δ h s
C is the jet velocity, Anozzle is the nozzle area, and ρtur,in is the density of the working fluid entering the turbine at the inlet. The relationship between the ideal efficiency of the turbine and the velocity ratio is as follows:
η i d e a l = 2 v 1 v 2
v = U C
U is the rotor tip speed.
In addition, introducing the relevant correction, the corrected efficiency formula is
η = η d e s i g n η i d e a l = η d e s i g n 2 v 1 v 2
In the compressor and turbine, the compression and expansion processes are non-isentropic. After obtaining the efficiency from the above processes, the exit enthalpy can be calculated using Equations (6) and (7):
h c o m , o u t = h c o m , i n + h c o m , o u t h c o m , i n η c o m
h t u r , o u t = h c o m , i n η t u r h t u r , i n h t u r , o u t
The work consumption can be calculated using Equations (8) and (9):
W c o m = m c o m h c o m , o u t h c o m , i n η c o m
W t u r = m t u r η t u r h t u r , i n h t u r , o u t
where h is the enthalpy, η is the efficiency, the subscript s represents the isentropic process, com represents the compressor, and tur represents the turbine.
The exit pressure of the turbine and compressor is a dynamic process that varies with mass flow rate, and the following equations describe it.
P o u t = P r e f P
P = m R T V M
where M is the molar mass, R is the gas constant, and T is the temperature. The differential form is as follows:
V d P d t + P d V d t = R M ( T d m d t + m d T d t )
When the volume and temperature do not change over a short period, the above equation can be further simplified as
d P d t = m ˙ T R M V
The thermal efficiency of the cycle is
η i = W t u r W M C W R C Q i n
where Q is the input heat, and the subscript in represents the heat source input power.
The flow rate in the compressor and turbine is calculated using the corrected Flugel formula as follows:
G c o m = g L C × P i n P r e f × T r e f T i n
G t u r = g L T × P r e f P i n × T i n T r e f

2.2.2. The Model Pre-Cooler and Heat Exchanger

The heat exchanger uses PCHE, and the dynamic equations for the heat exchange equipment are established based on the mass conservation and energy conservation equations. The mass conservation equations for the cold and hot fluids are as follows [26]:
V d ρ d t = m i n m o u t
The energy conservation equation for the hot fluid is as follows:
m ˙ h C p , h ( T h 1 T h 2 ) Q = ρ h V h C p , h d T ¯ h d t
The energy conservation equation for the cold fluid is as follows:
m ˙ c C p , c ( T c 1 T c 2 ) + Q = ρ c V c C p , c d T ¯ c d t
The average temperature of the heat exchanger is
T ¯ h = 1 2 ( T h 1 + T h 2 )
T ¯ c = 1 2 ( T c 1 + T c 2 )
The heat transfer formula for the heat exchanger is as follows:
Q = U 0 A T m
where ΔTm is the logarithmic mean temperature difference, and the calculation formula is as follows:
T m = ( T g T l ) I n ( T g T l )
where mh and mc are the mass flow rates of the hot and cold side fluids in the heat exchanger, respectively. Th1, Th2, Tc1, and Tc2 are the inlet and outlet temperatures of the hot and cold side fluids in the heat exchanger, respectively. Cp,h and Cp,c are the specific heat capacities of the hot and cold side fluids, respectively. Q is the heat transfer rate of the heat exchanger, Vh and Vc are the volumes of the hot and cold side fluids in the heat exchanger, t is time, and U0 and A are the overall heat transfer coefficient and the total heat transfer area, respectively.
Based on the above formula, its differential form can be written as follows:
d T h 2 d t = 2 m ˙ h ρ h V h ( T h 1 T h 2 ) 2 Q ρ h V h c p , h d T h 1 d t
d T c 2 d t = 2 m ˙ c ρ c V c ( T c 1 T c 2 ) + 2 Q ρ c V c c p , c d T c 1 d t
The pressure drop is calculated by Equation.
Δ p = Δ p f + Δ p l + Δ p a
where Δpf, Δpc, and Δpa are the friction pressure drop, form resistance pressure drop, and acceleration pressure drop, respectively.
Δ p f = f L D h ρ u 2 2
Δ p l = C ρ u 2 2
Δ p a = G 2 ( 1 ρ o u t 1 ρ i n )
f is the friction factor.
f = 15 . 78 R e R e < 2300 1 4 1 1.8 l o g ( R e ) 1.5 R e 2300
Based on the above formula, a calculation model was established in Simulink. Additionally, to construct the dynamic simulation, the complex differential equations required for the model were solved using Simulink’s S-Function module. During the calculation process, a variable-step solver was used, with the solver set to ode45s. The simulation accuracy was set to 0.1%, and the minimum step size was 0.001 s. Furthermore, the maximum step size was automatically selected by the solver. The thermodynamic properties of CO2 and its mixed working fluids were obtained by calling the NIST database, which provided detailed properties for CO2 and its mixed working fluids. The NIST database is based on a multi-parameter Helmholtz free energy equation of state and has been calibrated using experimental data, thus offering high accuracy and reliability.

3. Model Verification

3.1. Heat Exchanger Model Verification

To verify the correctness and accuracy of the heat exchanger model, the same model parameters as in [27] were used for modeling. A sinusoidal response was applied to the hot side inlet temperature, and the outlet temperature’s dynamic response was compared with Deng’s calculation results [27]. As shown in Figure 4, the relative error of the outlet temperatures on both sides is within 5%. In addition, the mean absolute errors of the outlet temperatures at the cold and hot sides of the heat exchanger were calculated, with values of 0.8679 K and 0.5951 K, respectively. These results validate the accuracy of the heat exchanger model used in this study.

3.2. The Verification of Cycle System

The same model data used by the Sandia Lab [28] were employed to verify the accuracy and validity of the established system. Key parameters at each critical point were calculated and compared with the results from the Sandia Lab [28]. As shown in Figure 5, the comparison between the results from this study and those from Sandia Lab indicates that the maximum error is less than 3%. In addition, the mean absolute errors of the temperature and pressure at key points were calculated, with values of 11.89 K and 0.076 MPa, respectively. These results validate the system’s accuracy established in this study, indicating that the system can be used for subsequent computational analysis.

4. Results and Discussion

The feasibility of using Kr as an additive in the sCO2 recompression Brayton cycle has already been proven [29]. Compared to pure CO2, using a CO2–Kr mixture working fluid improves the cycle’s energy efficiency. Additionally, the CO2–Kr mixture helps reduce exergy losses, especially by effectively controlling exergy destruction in the high-temperature heat exchanger, further optimizing the system overall performance. With the same exergy efficiency, the CO2–Kr cycle has a lower cost than the pure CO2 cycle, indicating that Kr as an additive improves cycle efficiency and provides better economic benefits [29]. However, in practical operation, the system will inevitably experience continuous fluctuations, and the dynamic performance of the CO2–Kr mixture as the working fluid remains unclear. Therefore, to predict and evaluate the dynamic response characteristics of the system under disturbance factors and explore the reasons for dynamic performance differences between different working fluids, a comparative analysis of the system’s dynamic response characteristics under mass flow, heat source power, and rotational speed disturbances was conducted.

4.1. Effects of Mass Flow Rate Step Change

Figure 6 shows the temperature response distribution at key locations in the CO2 and CO2–Kr systems under mass flow disturbance. When the system reaches steady operation, a downward step disturbance in mass flow is applied. The figure shows that, after the disturbance is applied, the temperature at key locations first decreases and then increases. This is because, with constant rotational speed and heat source power, a decrease in mass flow increases the heat absorbed per unit of working fluid, which causes the temperature to rise. However, in the early stages of the disturbance, due to the thermal inertia effect caused by the working fluid properties and the lag effect of the heat source, the temperature change at key locations initially shows a downward trend. Around 53 s, the thermal inertia and lag effects gradually disappear, and the temperature rises. Therefore, the final stable temperature at the turbine inlet and outlet is higher than the pre-disturbance temperature. For the re-compressor, during the final stable phase, the effect of the mass flow decrease on the re-compressor temperature is higher than the effect of efficiency decrease on the temperature. As a result, the temperature during the stable phase of the re-compressor is slightly lower than in the initial phase.
Taking the turbine outlet temperature as an example, a comparison of the response time and overshoot between the two working fluids reveals that, during the initial stage, the CO2–Kr system has a response time that is 3.5 s faster than the CO2 system. However, it also results in a larger response amplitude, with a value that exceeds CO2 by 6.25 K. After the disturbance is applied, in the turbine outlet temperature dynamic response, the pure CO2 system experiences a sharp drop in temperature followed by significant fluctuations (beginning to recover after 53 s), while the CO2–Kr mixed working fluid system shows a gentler fluctuation and a smooth decline, quickly recovering to the steady state. The response time is shortened by 10 s, and the final steady-state temperature is lower than that of the CO2 system. In the turbine inlet temperature response curve, the green shaded area clearly shows that the overshoot of the CO2–Kr system is much more pronounced than that of the CO2 system. Similarly, at the re-compressor inlet and outlet, the CO2–Kr system has a better temperature response time than the CO2 system, with a 1.5 s reduction in response time.
By comparing the temperature responses of the CO2 and CO2–Kr systems, it can be observed that the CO2-Kr mixture generally performs better dynamically after disturbances than the pure CO2 system. When faced with disturbances, the CO2 system experiences larger temperature fluctuations, longer recovery times, and a higher final stable temperature. In contrast, the CO2-Kr system exhibits smaller fluctuations, faster recovery, and a lower final temperature after the disturbance. This indicates that the addition of Kr improves the system’s stability and thermal efficiency, allowing the system to maintain smoother operation under disturbances.
Similarly, a decrease in mass flow leads to changes in the compression ratio of the compressor and turbine, which in turn causes a step change in the pressure at key locations. Figure 7 shows the pressure response at key locations. According to the compressor performance curve, underrated speed and a decrease in mass flow leads to an increase in the compression ratio, which increases the main compressor outlet pressure and turbine inlet pressure. However, the reduction in mass flow causes a decrease in the turbine compression ratio. In the early stages of the disturbance, the turbine outlet pressure decreases, followed by a decrease in the re-compressor inlet pressure. As the main compressor outlet pressure continues to increase, the system’s pressure gradually rises, leading to an increase in both the turbine outlet pressure and the re-compressor inlet pressure.
In addition, comparing the pressure response curves of CO2 and CO2–Kr, it is observed that both systems follow similar overall trends, but there are significant differences in overshoot and response time. Specifically, during the pressure response process, the overshoot for the pure CO2 system is 1.2 MPa, while the CO2–Kr system has an overshoot of only 0.5 MPa, with the difference being approximately 0.7 MPa. This difference indicates that the CO2–Kr mixture exhibits better stability during regulation and can effectively suppress excessive fluctuations. Regarding response time, under the same disturbance conditions, the CO2–Kr system’s response time is about 1.2 s shorter than that of the CO2 system. This reduction is approximately 16.7%, significantly improving the system’s dynamic performance. This improvement is attributed to the differences in thermophysical properties between CO2 and CO2–Kr, as the introduction of Kr enhances the gas’s density and viscosity, leading to lower overshoot and faster response speeds during pressure regulation. These properties enable the CO2–Kr system to reach a steady state more quickly and maintain smaller fluctuation amplitudes when facing disturbances.
During the system operation, the decrease in mass flow directly affects the turbine output power and compressor power consumption. As the mass flow decreases, the turbine output and the compressor power consumption reduce. However, the impact of the decrease in compressor power consumption is much greater than that of the turbine output power change. This is because the compressor power consumption has a more significant impact on the system’s overall energy consumption. Specifically, after the reduction in compressor power consumption, the overall system efficiency improves, leading to an increase in the net output power of the system. This change increases the system’s specific work and improves the thermal efficiency, provided that the heat source conditions remain unchanged, as shown in Figure 8.
By comparing the system performance of two different working fluids, it is observed that there are certain differences in turbine output power and compressor power consumption between the CO2 and CO2–Kr mixture systems. These differences mainly arise from the thermodynamic property differences of the working fluid at key locations, especially the differences in physical parameters such as density and viscosity under different operating conditions. The CO2 system has a specific work of 42.52 kW/kg and a thermal efficiency of 30.37% after the disturbance. In comparison, the CO2–Kr system demonstrates better performance with a specific work of 50.39 kW/kg and a thermal efficiency of 35.21%. These results indicate that the CO2–Kr mixture performs better than the pure CO2 system, particularly in specific work and thermal efficiency, showing a clear advantage.

4.2. Effects of Heat Source Step Change

When the system operates stably at the design operating point, a heat source disturbance is applied, reducing the heat source from 810 kW to 730 kW. Figure 9 shows the dynamic temperature response of the system after the heat source disturbance. It can be observed that, during the 50 s heat source decrease, the temperature response curves at the inlet and outlet of the re-compressor and turbine exhibit significant dynamic changes. For the CO2 working fluid, the re-compressor inlet temperature stabilizes at 353 K before the disturbance. However, a sudden change occurs after the heat source decreases, and the temperature drops sharply to 344 K at 50.4 s, a decrease of approximately 2.5%. Then, the temperature gradually rises to 346.7 K at 50.8 s, entering a state of slight fluctuation. The turbine inlet temperature initially stabilizes at 828 K before the disturbance. Following the decrease in heat source, the turbine inlet temperature drops rapidly to 822.08 K at 50.4 s. Similarly, the turbine outlet temperature also exhibits a decreasing trend.
This overall trend is primarily driven by the thermodynamic response mechanism of the system caused by decrease in the heat source. The reduction in heat source power directly leads to a decrease in the thermal load of the cycle system. The re-compressor needs to adjust its compression ratio to maintain flow balance, causing the inlet temperature to fluctuate due to changes in the working fluid’s heat absorption capacity. The brief rise in turbine inlet temperature results from the delayed response due to the system’s thermal inertia, followed by a gradual decrease as the heat source disruption continues.
In contrast, the CO2–Kr mixture shows a more gradual temperature change. The re-compressor inlet temperature, initially stable at 352.83 K, drops to 341.94 K at 50.4 s after the heat source disturbance, with a decrease of approximately 3.1%, slightly higher than pure CO2. The turbine inlet temperature is 828.06 K at 50 s, which is 2.07 K lower than that of CO2, and the subsequent fluctuation is smaller. Notably, the difference in turbine outlet temperature between the two working fluids is more significant. At 50 s, the CO2–Kr outlet temperature is 767.07 K, 1.53 K lower than CO2’s 765.44 K. This indicates that the mixture has better thermal stability.
These differences can be attributed to the inert gas properties of Kr. Its larger molecular weight alters the working fluid’s heat capacity and thermal diffusion resistance [16]. After a disturbance occurs, CO2–Kr, with its lower specific heat capacity and thermal conductivity, experiences relatively slow heat transfer, resulting in smaller temperature fluctuations from the disturbance, which effectively reduces system oscillations.
Similarly, the pressure response curves at the inlet and outlet of the re-compressor and turbine show dynamic changes, with significant differences between the CO2 and CO2–Kr working fluids, as shown in Figure 10. For the CO2, the re-compressor inlet pressure gradually rises from the initial value of 7.75 MPa to 7.90 MPa at 50 s. However, after the heat source decreases, the inlet pressure drops only slightly and then further decreases to 7.895 MPa at 50.7 s, accompanied by small fluctuations. The outlet pressure slowly increases from the initial 13.73 MPa to 13.91 MPa at 50 s, and after the disturbance, it gradually stabilizes at 13.81 MPa, with a small overall fluctuation. The turbine inlet pressure in the CO2 system gradually rises from 13.49 MPa to 13.68 MPa at 50 s and then continues to fluctuate, reaching 13.569 MPa at 100 s. The turbine outlet pressure rises from 7.89 MPa to 8.04 MPa at 50 s, then quickly drops to 8.00 MPa after the heat source decreases, finally stabilizing at 7.93 MPa at 100 s. This trend is primarily driven by the thermodynamic imbalance caused by the sudden drop in heat source power. The reduced thermal load leads to a decline in the heat absorption capacity of the working fluid. To maintain flow balance, the re-compressor needs to adjust its compression ratio, which causes fluctuations in the inlet pressure due to changes in the fluid density. The brief increase in turbine inlet pressure results from the system’s inertia, causing a delayed response and a gradual decrease due to the interruption of the heat source.
In contrast, the pressure variations in the CO2–Kr mixture are more gradual. The re-compressor inlet pressure increases from 7.75 MPa to 7.89 MPa at 50 s, then decreases to 7.85 MPa at 50.3 s after the heat source decreases, with a decrease of only 0.04 MPa. The turbine inlet pressure at 50 s is 13.65 MPa, which is 0.03 MPa lower than that of CO2, and during subsequent fluctuations, the maximum decrease is 0.12 MPa. The difference in turbine outlet pressure is more noticeable, with CO2–Kr at 50 s being 8.041 MPa, 0.001 MPa lower than CO2. However, at 100 s, it stabilizes at 7.92 MPa, which is 0.01 MPa lower than CO2’s 7.93 MPa, indicating that the CO2–Kr mixture has superior long-term stability. This difference is mainly due to the distinct thermodynamic properties of the two working fluids. As a single fluid, CO2 shows significant pressure changes when the heat source fluctuates. However, introducing the additive Kr alters the thermodynamic characteristics of the system. Kr’s low-temperature properties make the CO2–Kr system more stable during heat source fluctuations. Kr optimizes the thermodynamic performance of the gas mixture, leading to smaller pressure fluctuations and faster responses when the heat source changes in the CO2–Kr system.
Due to the impact of the aforementioned key parameters, the system’s thermal performance changes, as shown in Figure 11, which illustrates the system’s dynamic performance response curve during heat source disturbances. In the CO2 system, before the heat source disturbance (0–50 s), the efficiency and specific work stabilized at 28.47% and 39.49 kW/kg, respectively. At 50.3 s, after the sudden drop in the heat source, the efficiency sharply rises to 29.70%, while the specific work drops to 37.64 kW/kg. Subsequently, at 50.7 s, the efficiency further increases to 30.20%, a 1.73% improvement from the initial value, and the specific work rebounds to 38.25 kW/kg, gradually stabilizing in subsequent fluctuations. Finally, the efficiency stabilizes at 31.24%, and the specific work stabilizes at 39.52 kW/kg, reflecting an increase of 2.77% and 0.03 kW/kg, respectively, from the initial values.
This change is due to the reduction in heat load, which causes the turbine output power to decrease. The system achieves a new energy balance, leading to a stable state. The rise in efficiency is primarily driven by the positive effect of reduced compressor power consumption, which outweighs the negative effect of reduced turbine work, leading to the overall efficiency increase. Similarly, the specific work shows a similar trend.
In contrast, the efficiency and specific work of the CO2–Kr system are significantly higher than those of the CO2 system before the heat source decreases. After the heat source decreases, the efficiency of the CO2–Kr system drops to 35.23% at 50.3 s, and the specific work drops to 44.58 kW/kg. However, the recovery is faster. By 51.5 s, the efficiency rebounds to 36.11%, and the specific work returns to 45.68 kW/kg. By 100 s, the efficiency stabilizes at 37.01%, 5.77% higher than CO2, and the specific work stabilizes at 46.81 kW/kg, 7.29 kW/kg higher than CO2. The above results show that the CO2–Kr system exhibits less fluctuation in efficiency during heat source disturbances and reaches a stable specific work value more quickly, demonstrating superior disturbance resistance. The larger molecular weight of Kr leads to a lower specific heat capacity of the CO2–Kr mixture, allowing the mixture to respond more quickly to temperature disturbances and recover to a steady state faster. This helps reduce the impact of temperature fluctuations on system performance when there are sudden changes in the heat source. Furthermore, CO2-Kr’s viscosity is higher than pure CO2, which increases flow resistance but reduces compressor power consumption, thus making efficiency improvements more pronounced.
In summary, CO2–Kr, due to its optimized thermodynamic properties—specifically, the increase in density—significantly reduces the actual volumetric flow rate compared to pure CO2. The reduction in volumetric flow lowers the flow velocity of the working fluid in pipes and equipment, thus minimizing the amplification effect of flow inertia on disturbances. Under dynamic operating conditions, in the CO2–Kr mixture, the addition of Kr moderately increases the mixture’s viscosity compared to pure CO2, thereby suppressing flow instabilities, while Kr’s inertness improves thermodynamic stability under high-temperature conditions. Although Kr’s thermal conductivity is slightly lower than CO2 at standard conditions, the mixture’s enhanced density and adjusted transport properties promote more efficient heat transfer in certain regimes. These combined effects contribute to an improvement in isentropic efficiency during compression/expansion and significantly dampen pressure fluctuations.

4.3. Effects of Speed Step Change

In the recompression Brayton cycle system, the compressor and turbine are arranged with a single shaft, and disturbances in the rotational speed affect the performance curves of the compressor and turbine, which in turn impact the system’s key parameters and, ultimately, the system performance. The temperature response characteristics of the two working fluids are shown in Figure 12. For the CO2 system, the re-compressor inlet temperature gradually rises from 335.20 K at 0.2 s to 353.20 K just before 50 s. After a speed decrease at 50.2 s, it sharply increases to 353.20 K then rises to 355.99 K by 50.6 s, with small fluctuations thereafter, stabilizing at 354.17 K by 100 s. The outlet temperature rises from 389.95 K to 411.81 K just before 50 s. After the speed decreases, it reaches 415.13 K at 50.6 s, eventually stabilizing at 412.91 K. The turbine inlet temperature increases from 815.99 K to 829.20 K just before 50 s, and after the speed decreases, it jumps to 831.28 K at 50.2 s, fluctuating to 829.20 K by 100 s. The turbine outlet temperature rises from 754.40 K to 769.81 K just before 50 s. After the speed decrease, it reaches 771.12 K at 50.6 s, finally stabilizing at 768.82 K. This trend is mainly caused by the thermodynamic inertia effects triggered by the rotational speed decrease. The speed decrease causes an imbalance in the power output of the compressor and turbine. The reduction in the compressor compression ratio lowers the temperature rise of the working fluid, while the weakening turbine power output delays heat expulsion, leading to instantaneous temperature accumulation. In contrast, the CO2–Kr mixture exhibits better temperature stability from the beginning. Specifically, the re-compressor inlet temperature rises from 344.17 K to 352.83 K before 50 s, slightly lower than the CO2 system. After the speed decrease, the peak temperature reaches 359.47 K, finally stabilizing at 355.17 K. The turbine inlet temperature starts at 842.24 K, and the fluctuation is only 0.6% after the speed decrease, eventually stabilizing at 829.78 K. The superiority of CO2–Kr stems from its physical property parameters, as the specific heat capacity of CO2–Kr is lower than that of pure CO2. This allows the CO2–Kr system to respond more quickly to disturbances and recover to a steady state faster, thereby suppressing temperature fluctuations during sudden changes in rotational speed. Therefore, CO2–Kr mitigates temperature fluctuations through its lower specific heat capacity, demonstrating better performance under dynamic conditions. It provides a reliable working fluid choice for recompression Brayton cycle systems in high fluctuation scenarios.
During the speed drop disturbance, the pressure response characteristics of CO2 and CO2–Kr show significant differences, as shown in Figure 13. After the speed disturbance occurs, the system enters the dynamic adjustment phase. Taking CO2 as an example, the re-compressor inlet pressure gradually rises from 7.757 MPa to 7.89 MPa, while the turbine outlet pressure rises from 7.897 MPa to 8.04 MPa. The speed drop directly leads to a reduction in the compressor power consumption and a decrease in the compression ratio. It thus causes a redistribution of the working fluid’s flow and pressure. In contrast, the CO2–Kr re-compressor inlet pressure rises from 7.758 MPa to 7.922 MPa, and the turbine outlet pressure rises from 7.898 MPa to 8.040 MPa. This indicates that the pressure response of CO2 is more intense at the initial stage of the speed drop, while the pressure change of CO2–Kr is relatively more gradual. However, both systems show an increasing pressure trend during the transition phase before reaching a steady state. It is worth noting that the change in turbine inlet pressure is more pronounced. In the CO2 system, the pressure rises from 13.730 MPa to 13.682 MPa, while in CO2–Kr, it drops from 13.730 MPa to 13.676 MPa. This difference may be due to the thermodynamic properties of the working fluid affecting the turbine’s energy conversion efficiency.
Similarly, the system performance exhibits corresponding response characteristics after the disturbance, as shown in Figure 14. The efficiency and specific work response curves of CO2 and CO2–Kr show significant differences. During the initial stable phase, the efficiency and specific work of CO2–Kr are significantly higher than those of CO2. As the speed decreases, both working fluids exhibit a decreasing trend in efficiency and specific work, but the rate of decrease and the final stable values differ. At 50 s, the efficiency of CO2 drops to 28.48%, a decrease of 0.74% from the initial value, while the efficiency of CO2–Kr drops to 33.76%, a decrease of 1.17%. Regarding specific work, CO2 drops from 39.51 kW/kg to 38.49 kW/kg, while CO2–Kr drops from 47.29 kW/kg to 45.64 kW/kg. Between 50 and 100 s, the efficiency and specific work of CO2 tend to stabilize. The efficiency and specific work of CO2–Kr also stabilize, but the drop is more significant, and the values remain consistently higher than those of CO2.
This is related to the thermodynamic properties of the working fluids and the system’s dynamic response. CO2–Kr performs excellently during the initial high-speed phase due to its heat capacity and better compressibility, which allows for more efficient energy transfer under high-pressure and high-temperature conditions, thereby improving cycle efficiency and output power. However, as the speed decreases and the system enters a low-load state, the working fluid’s flow resistance and heat losses gradually become dominant. CO2, with its lower viscosity and higher thermal diffusivity, can maintain more stable flow characteristics at low speeds, thus slowing down the rate of decrease in efficiency and specific work. In contrast, the thermophysical properties of CO2–Kr are more sensitive to flow rate changes, leading to a more significant performance degradation.

4.4. Analysis of Working Fluid Property Differences

The above analysis discussed the dynamic performance of CO2 and CO2–Kr as working fluids under different disturbance factors. In this section, a detailed analysis will be conducted from the perspective of the working fluid properties, providing a theoretical basis for system optimization and operational control.
Figure 15 compares the density, viscosity, specific heat capacity, and thermal conductivity of two working fluids, CO2 and CO2–Kr, within the temperature range of 300 K–900 K and pressure range of 5 MPa–30 MPa. The figure shows that the CO2–Kr mixture exhibits significant differences from pure CO2 in terms of density, viscosity, specific heat capacity, and thermal conductivity. The density of the CO2–Kr mixture is higher than that of pure CO2. As the temperature increases, the density decreases more significantly, indicating that its volume expansion effect is more pronounced under high-temperature conditions. CO2–Kr generally has a higher viscosity than pure CO2, with the difference becoming more pronounced at higher temperatures and pressures. This results in lower friction losses in the CO2–Kr system, thus improving system efficiency. Regarding specific heat capacity, the CO2–Kr mixture has a lower specific heat capacity than pure CO2, especially at high temperatures and low pressures. This means that the CO2–Kr system has weaker heat absorption and release capabilities, allowing for a faster response to temperature changes but with a relatively weaker thermal buffering capacity. The CO2–Kr mixture also has a lower thermal conductivity than pure CO2, indicating weaker heat transfer capabilities, which may slow heat transfer during thermal regulation, thus affecting the system’s thermal efficiency and dynamic response characteristics.
In summary, the density of pure CO2 is lower than that of the CO2–Kr mixture and decreases more significantly with increasing temperature. This is due to the introduction of Kr, which significantly increases the working fluid’s molecular weight and enhances the thermal inertia effect. Its inert gas properties cause the mixture to exhibit a delayed temperature response during the initial phase of dynamic disturbances. In contrast, the more pronounced volume expansion effect at high temperatures results from changes in intermolecular forces. The pure CO2 generally has a lower viscosity than the CO2–Kr mixture, with this difference being more significant under high temperature and pressure conditions. This allows the mixture to suppress turbulent fluctuations and reduce the amplitude of pressure fluctuations. Its optimized compressibility enables pressure disturbances to be more efficiently transmitted downstream and reduces pressure oscillations through enhanced damping effects, thereby shortening the system’s response time.
Regarding thermodynamic properties, the mixture’s specific heat capacity and thermal conductivity are lower than those of the pure CO2 system. While this weakens the heat absorption and conduction capabilities, it restructures the transient energy transfer mechanism. The lower specific heat capacity makes the CO2–Kr system more sensitive to temperature changes. At the same time, the decrease in thermal conductivity, in conjunction with the viscosity and density, forms a unique thermal buffering mechanism. During the later stages of the disturbance, the system’s higher thermal capacity effectively dampens temperature fluctuations. The coupling effects of these physical properties enable the CO2–Kr mixture to exhibit superior thermal stability during dynamic disturbances, thereby enhancing the dynamic response capability and disturbance resistance of the Brayton cycle system.

5. Conclusions

This paper establishes a dynamic model of the re-compression Brayton cycle using CO2 and CO2–Kr mixtures based on the performance curves of the compressor and turbine and the differential equations of the heat exchange equipment. The dynamic response characteristics of CO2 and CO2–Kr mixtures under different disturbance factors are thoroughly discussed, and the reasons for the dynamic performance differences are analyzed from the perspective of thermophysical property coupling. The main conclusions are as follows.
(1)
Under mass flow disturbances, the dynamic performance of the CO2–Kr mixture is superior to that of the pure CO2 system. Specifically, the turbine outlet temperature response time of the CO2–Kr system is 3.5 s faster than that of the CO2 system, and the response amplitude is 6.25 K lower, with smaller temperature fluctuations. The CO2 system experiences significant temperature fluctuations after the disturbance, and recovery begins only after 53 s, while the CO2–Kr system exhibits a smooth decrease and quickly recovers to a stable state, with a response time reduced by 10 s. The final temperature of the CO2–Kr system is lower than that of the CO2 system. Similarly, in the turbine inlet and re-compressor temperature responses, the CO2–Kr system’s response time is also shortened by 1.5 s, demonstrating its better thermal efficiency and stability.
(2)
When the heat source power drops from 810 kW to 730 kW, the inlet temperature of the re-compressor in the CO2 system decreases to 344 K (a 2.5% reduction). In contrast, the CO2–Kr system experiences a 3.1% reduction, with smaller temperature fluctuations. The pressure fluctuations in the CO2 system are larger, whereas the CO2–Kr system is more stable. In terms of efficiency and specific work, the CO2 system sees a 2.77% increase in efficiency, with a slight recovery in specific work. The CO2–Kr system, on the other hand, achieves an efficiency of 37.01% and a specific work of 46.81 kW/kg, which are higher than the CO2 system by 5.77% and 7.29 kW/kg, respectively. The advantage of the CO2–Kr mixture comes from the optimized thermophysical properties of Kr, which improve the system’s thermal stability and energy conversion efficiency.
(3)
When a speed disturbance occurs, the inlet temperature of the re-compressor in the CO2 system rises from 335.20 K to 355.99 K, and the inlet temperature of the turbine increases from 815.99 K to 831.28 K. In contrast, the temperature fluctuations in the CO2–Kr system are smaller, stabilizing at 355.17 K and 829.78 K. The pressure variations in the CO2–Kr system are smaller, stabilizing at 7.922 MPa and 8.04 MPa. In terms of efficiency and specific work, the efficiency of the CO2 system drops to 28.48%, and the specific work decreases to 38.49 kW/kg. Meanwhile, the efficiency of the CO2–Kr system decreases to 33.76%, and the specific work drops to 45.64 kW/kg. Overall, the CO2–Kr system demonstrates higher stability.
(4)
The differences in dynamic response between the CO2 and CO2–Kr mixed working fluid systems arise from the property changes brought about by introducing Kr. Kr alters the heat capacity, thermal conductivity, and viscosity of the mixed fluid, enhancing thermal inertia, which helps buffer temperature fluctuations and reduces the impact of transient thermal shocks on the system. At the same time, the improved thermal conductivity and viscosity optimize heat transfer and flow stability, reducing pressure fluctuations. This enables the CO2–Kr system to demonstrate superior thermal stability and disturbance resistance under dynamic disturbances, thus improving the dynamic performance of the recompression Brayton cycle.
In this study, it was found that the CO2–Kr mixture exhibits better dynamic response characteristics compared to pure CO2, especially in temperature and pressure regulation, where it shows lower overshoot and faster recovery. These characteristics provide important insights for the control system design of Brayton cycle systems, allowing for adjustments in control algorithms to improve system stability and response speed. Furthermore, the CO2–Kr mixture optimizes the system’s thermodynamic performance and disturbance resistance, particularly under high load fluctuations or sudden changes, effectively mitigating temperature fluctuations and enhancing overall efficiency. In future practical applications, Kr could be considered as an additive, added in a specific ratio, to achieve stable and efficient system operation.

Author Contributions

Conceptualization, L.C.; Methodology, M.F.; Investigation, M.F.; Writing—original draft, M.F.; Writing—review & editing, X.X. and Q.M.; Visualization, M.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. Schematic diagram and T-s diagram of the compression Brayton cycle. (a) Schematic diagram of the operation of the recompression cycle system; (b) T-s diagram of the recompression Brayton cycle.
Figure 1. Schematic diagram and T-s diagram of the compression Brayton cycle. (a) Schematic diagram of the operation of the recompression cycle system; (b) T-s diagram of the recompression Brayton cycle.
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Figure 2. The characteristic curves of the compressor. (a) The pressure ratio curve of the compressor; (b) The efficiency curve of the compressor.
Figure 2. The characteristic curves of the compressor. (a) The pressure ratio curve of the compressor; (b) The efficiency curve of the compressor.
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Figure 3. The characteristic curves of the turbine. (a) The pressure ratio curve of the turbine; (b) The efficiency curve of the turbine.
Figure 3. The characteristic curves of the turbine. (a) The pressure ratio curve of the turbine; (b) The efficiency curve of the turbine.
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Figure 4. The verification of PCHE [27].
Figure 4. The verification of PCHE [27].
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Figure 5. Comparison of calculation results with experimental value [28].
Figure 5. Comparison of calculation results with experimental value [28].
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Figure 6. The temperature response at key positions under mass flow disturbances.
Figure 6. The temperature response at key positions under mass flow disturbances.
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Figure 7. The pressure response at key positions under mass flow disturbances.
Figure 7. The pressure response at key positions under mass flow disturbances.
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Figure 8. The system performance response at key positions under mass flow disturbances.
Figure 8. The system performance response at key positions under mass flow disturbances.
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Figure 9. The temperature response at key positions under heat source disturbances.
Figure 9. The temperature response at key positions under heat source disturbances.
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Figure 10. The pressure response at key positions under heat source disturbances.
Figure 10. The pressure response at key positions under heat source disturbances.
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Figure 11. The system performance response at key positions under heat source disturbances.
Figure 11. The system performance response at key positions under heat source disturbances.
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Figure 12. The temperature response at key positions under speed disturbances.
Figure 12. The temperature response at key positions under speed disturbances.
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Figure 13. The pressure response at key positions under speed disturbances.
Figure 13. The pressure response at key positions under speed disturbances.
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Figure 14. The system performance response at key positions under speed disturbances.
Figure 14. The system performance response at key positions under speed disturbances.
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Figure 15. Comparison of physical properties among different working media. (a) Density; (b) Viscosity; (c) Specific heat capacity; (d) Thermal conductivity coefficient.
Figure 15. Comparison of physical properties among different working media. (a) Density; (b) Viscosity; (c) Specific heat capacity; (d) Thermal conductivity coefficient.
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Fang, M.; Cao, L.; Xu, X.; Meng, Q. The Dynamic Response Characteristics and Working Fluid Property Differences Analysis of CO2–Kr Mixture Power Cycle System. Processes 2025, 13, 1735. https://doi.org/10.3390/pr13061735

AMA Style

Fang M, Cao L, Xu X, Meng Q. The Dynamic Response Characteristics and Working Fluid Property Differences Analysis of CO2–Kr Mixture Power Cycle System. Processes. 2025; 13(6):1735. https://doi.org/10.3390/pr13061735

Chicago/Turabian Style

Fang, Minghui, Lihua Cao, Xueyan Xu, and Qingqiang Meng. 2025. "The Dynamic Response Characteristics and Working Fluid Property Differences Analysis of CO2–Kr Mixture Power Cycle System" Processes 13, no. 6: 1735. https://doi.org/10.3390/pr13061735

APA Style

Fang, M., Cao, L., Xu, X., & Meng, Q. (2025). The Dynamic Response Characteristics and Working Fluid Property Differences Analysis of CO2–Kr Mixture Power Cycle System. Processes, 13(6), 1735. https://doi.org/10.3390/pr13061735

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