A carbon dioxide energy storage system with high-temperature graded heat storage structure: Thermodynamic intrinsic cycle construction and performance analysis

Abstract

Carbon dioxide energy storage (CES) is an emerging compressed gas energy storage technology which offers high energy storage efficiency, flexibility in location, and low overall costs. This study focuses on a CES system that incorporates a high-temperature graded heat storage structure, utilizing multiple heat exchange working fluids. Unlike traditional CES systems that utilize a single thermal storage at low to medium temperatures, this system significantly optimizes the heat transfer performance of the system, thereby improving its cycle efficiency. Under typical design conditions, the round-trip efficiency of the system is found to be 76.4%, with an output power of 334 kW/(kg·s^‒1) per unit mass flow rate, through mathematical modeling. Performance analysis shows that increasing the total pressure ratio, reducing the heat transfer temperature difference, improving the heat exchanger efficiency, and lowering the ambient temperature can enhance cycle efficiency. Additionally, this paper proposes a universal and theoretical CES thermodynamic intrinsic cycle construction method and performance prediction evaluation method for CES systems, providing a more standardized and accurate approach for optimizing CES system design.

Graphical abstract

Keywords

carbon dioxide energy storage (CES) / high-temperature graded heat storage / thermodynamic intrinsical cycle construction / performance analysis

1 Introduction

New energy storage plays a crucial role in modern power and new energy systems, and is a key technology in the pursuit of carbon neutrality [1]. As the installed capacity of new energy sources continues to grow, large-scale, long-term energy storage has increasingly become a social consensus. Traditional technologies such as pumped hydro storage (PHS), compressed air energy storage (CAES), and flow battery (FB), which have already been implemented, still face challenges such as long construction cycles [2], high geographical requirements [3], immature technologies [4], and limited adaptability [5]. In response, carbon dioxide energy storage (CES) has emerged as a promising large-scale energy storage technology. Based on thermodynamic Carnot and Brayton cycles, CES converts, stores, and releases electrical energy into internal energy through carbon dioxide and thermal storage media. CES can achieve large capacity, long-term energy storage, offering several advantages, including high cycle efficiency, no geographical limitations, intrinsic safety, and power capacity decoupling [6–8]. It holds great potential in large-scale energy storage applications such as new energy generation bases, centralized energy storage centers in power grids, and zero-carbon industrial parks.

Fig.1 shows the phase diagram of carbon dioxide. Compared to air, carbon dioxide has a lower critical point (7.38 MPa, 30.98 °C), which makes it easier to reach its supercritical state through artificial operation. This characteristic allows liquefy carbon dioxide to liquefy at ambient temperatures. Additionally, the density of liquid carbon dioxide is much higher than that of air at the same pressure, effectively reducing the storage volume needed for high-pressure working fluids [9]. Moreover, carbon dioxide in a supercritical state also has low viscosity, high specific heat capacity, and high density, making it an excellent thermodynamic working fluid with good fluidity and heat transfer efficiency [10].

Currently, much of the research on CES focuses on thermal electric CES [11,12] (TE-CES, also known as the carbon dioxide Carnot battery) and gas–liquid transcritical CES [13–15] (G-L-TCES). TE-CES only uses carbon dioxide as the circulating working fluid. In this system, electrical energy is converted into high-temperature thermal energy and low-temperature cold energy through the reverse Carnot cycle. The stored thermal and cold energy, along with some electrical energy is then used to drive a thermal cycle that generates electricity. With different settings of pressure and temperature parameters, TE-CES can also combine Rankine and Brayton cycles to enhance its performance. The electric-to-electric conversion efficiency of TE-CES generally ranges from 20% to 50%, but considering the energy savings from heat pumps and high-grade heat sources, its overall efficiency can exceed 1.0 [16], offering significant potential for multi-energy cogeneration systems.

Given the high demand of the power grid for the energy storage efficiency, this study primarily considers G-L-TCES. As shown in Fig.2, the main feature of the G-L-TCES system is that carbon dioxide serves both as the working medium for energy storage and release and as the medium for converting electrical energy into thermodynamic energy. The system includes low-pressure gas storage and high-pressure liquid storage devices, with carbon dioxide undergoing transcritical compression and expansion processes. G-L-TCES is equivalent to a general CES technology that uses ideal adiabatic compression and expansion. By achieving higher heat storage temperatures and appropriate liquefaction pressures with a larger pressure ratio, the system improves the power performance of the turbine, resulting in electrical efficiencies of the system generally exceeding 60%. Moreover, G-L-TCES can store high-density liquid carbon dioxide at or near room temperature without requiring large-scale high-pressure gas storage chambers. Therefore, G-L-TCES is experiencing rapid development.

Scholars have conducted extensive thermodynamic studies on CES systems to explore ways to improve system cycle efficiency through parameter optimization. Liu [17] classifies CES into transcritical and supercritical types, based on the state of carbon dioxide at the outlet of the expander. Thermodynamic analysis reveals that their cycle efficiencies under typical operating conditions are 54.43% and 53.02%, respectively. However, the energy storage density of the transcritical cycle is higher. Lu et al. [18] developed a high-pressure CES system based on underground storage and optimized it using a non-equal pressure ratio design method, achieving a cycle efficiency of up to 74%. Cao et al. [19] found that systems with multiple compression stages and fewer expansion stages have better cycle efficiency. While fewer compression stages lead to higher power consumption, they also allow for more compression heat recovery, which is more conducive to increasing the output power of the expander. Researchers have also worked on improving CES cycles. Zheng et al. [20] analyzed CES systems with different energy storage states and found that a system storing both high and low-pressure liquid phases had the best overall performance after considering economic investment. Hao et al. [21] designed a combined heat and power (CCES-CHP) system based on compressed CES and proposed an operational feasibility analysis method, providing a way to evaluate the cogeneration capacity and flexibility of the system.

To further investigate the key factors affecting the performance of the CES system, Tab.1 provides a summary of some of the CES system parameters from the literature. Preliminary conclusions can be drawn as follows ① CES systems with both low-pressure and high-pressure liquid storage have lower electrical efficiency; ② the number of compression stages directly affects the turbine inlet temperature, and higher temperatures lead to higher the electrical efficiency; ③ increasing storage pressure excessively does not necessarily improve electrical efficiency; ④ there is insufficient research on thermal storage in CES systems, with most studies using only a single thermal storage medium (such as pressurized water, thermal oil, or molten salt). Therefore, more focus is needed on optimizing the pressure, thermal storage temperature, thermal storage medium, and other parameters of the CES system.

In summary, the operation of CES systems involves multiple thermodynamic conversion processes, which differ significantly from conventional heat engine or heat pump cycles. However, existing studies have only focused on optimizing certain performance parameters of different CES systems, leaving the thermodynamic intrinsic mechanism of CES still unclear. There is a lack of clear thermodynamic boundary descriptions for key parameters such as heat storage temperature and cycle efficiency. As a result, the design of the CES system often lacks a structured approach and appears to be disorganized.

To address these gaps, this paper proposes an independent graded thermal storage CES system, along with a method for constructing a thermodynamic intrinsic cycle based on irreversible loss corrections. It also conducts an energy storage efficiency prediction analysis. Furthermore, it analyzed the internal energy conversion and parameter coupling relationship of the proposed system from the perspective of practical engineering design, and investigated the influence of key parameters on system performance. The schematic overview of the entire methodology is shown in Fig.3.

2 System description

As shown in Fig.4, this paper proposes a CES system with a high-temperature graded heat storage structure. This system is composed of several key components, including low-pressure energy storage units, power units (compressors and turbines), first to fourth stage heat storage units, and high-pressure energy storage units. Due to the wide thermal storage temperature range exceeding 400 K in the system and the varying optimal working temperature zones for each thermal storage medium, it is not feasible to use just one or two thermal storage media. To address this, molten salt (solar salt, a 60% NaNO₃ + 40% KNO₃ mixture), thermal oil (Therminal 55), pressurized water, and atmospheric water are used as thermal storage materials. Tab.2 presents the thermal properties and economic costs of the selected heat storage materials.

The advantage of this design is that it minimizes the use of high-cost materials while reducing heat transfer losses by appropriately dividing the heat storage temperature zone. The working principle of the system is divided into two distinct processes.

Energy storage process: CO₂ in the low-pressure storage (LPV) enters the compressor and is compressed to a supercritical state (or near the critical pressure), achieving high temperature and high pressure. The generated compression heat is transferred to the heat storage medium via heat exchangers at each stage of heat storage unit. The cooled CO₂ is then stored in the high-pressure storage tank (HPV).

Energy release process: High-pressure CO₂ is released from the HPV and enters the heat storage units at various stages to absorb heat and reheat. After this, it passes through the turbine to do work, driving the generator to produce electrical energy. The CO₂ discharged from the turbine is adjusted to its proper temperature and returned to the LPV for reuse.

From the perspective of innovation and improvement, the system increases the maximum hot side temperature achievable by employing a single-stage wide pressure differential compression and expansion design. According to existing research, this design can effectively increase the heat storage temperature of the system, which has a positive effect on improving overall system efficiency. Additionally, the use of graded heat storage minimizes heat loss that typically occurs in single-stage systems or systems using a single working fluid, further optimizing the performance of the system.

3 Thermodynamic intrinsical cycle construction and analysis

In the ideal Carnot cycle, it is assumed that heat exchange between the “heat engine” and the high- and low-temperature heat sources is transient, stable, non-dissipative, and fully reversible. However, in reality, heat exchange always involves irreversible factors. Extensions of classical thermodynamics have always tended to model real systems in a way that is closer to reality, especially the hard-to-ignore heat transfer losses, heat leakage, and friction losses [30]. Curzon et al. [31] hypothesized an internally reversible Carnot cycle that only has a heat transfer temperature difference with the external heat source, but remains completely reversible within the system. They derived an expression for its thermodynamic cycle efficiency as η_{thermodynamic, irreversible} = 1−(T_L/T_H) ^1/2, which is lower than the ideal Carnot cycle efficiency, thus limiting the maximum cycle efficiency achievable by traditional heat engines under the influence of external reversibility.

Fig.5 illustrates the basic thermodynamic process of the CES system with single-stage compression and expansion, as proposed in this study. The process is broken down into the following stages:

1‒2: Adiabatic Compression Process: CO₂ is compressed adiabatically, T₀ is the initial ambient temperature, T_H is the highest temperature of compressed exhaust in the system;

2‒3: Isobaric Heat Transfer Process: In this stage, CO₂ transfers heat to the heat storage medium and is cooled. T_HPV is the storage temperature. If the temperature difference and heat loss during the cooling process are ignored, the temperature of CO₂ can be considered to return to the ambient temperature T₀;

3‒4: Isobaric Heat Transfer Process: CO₂ absorbs and stores heat during this stage, which reheats it. However, due to the temperature difference between the cooler and heater, as well as the time scale of heat transfer, CO₂ can only recover to the turbine inlet temperature T_T;

4‒5: Adiabatic Expansion Process: Similar to the compression process, and due to isobaric nature, the outlet temperature T_L of the turbine may be lower than T₀;

5‒1: Natural Heating Process: After the working cycle of the CES system ends, natural heating occurs to bring the system back to its initial state.

Furthermore, it can be considered that Stages 1‒2 and 2‒3 are the half inverse Brayton cycle, while 3‒4 and 4‒5 are the half positive Brayton cycle. The combination of 1‒2‒4‒5 forms a new type of inverse Brayton cycle.

(1)w_{\mathrm{C}}=s_{\mathrm{C}}(T_{\mathrm{H}}- T_0),

(2)w_{\mathrm{T}}=s_{\mathrm{T}}(T_{\mathrm{T}}- T_{\mathrm{L}}),

(3)\mathrm{\Delta }{s}_{\mathrm{H}}= s_{\mathrm{C}}-s_{\mathrm{T}}= c{}_{\mathrm{p}}\mathrm{l}\mathrm{n}\frac{T_{\mathrm{H}}}{T_{\mathrm{T}}},

(4)\mathrm{\Delta }{s}_{\mathrm{L}}= s_{\mathrm{C}}-s_{\mathrm{T}}= c{}_{\mathrm{p}}\mathrm{l}\mathrm{n}\frac{T_0}{T_{\mathrm{L}}},

where w_{\text{C}} and w_{\text{T}} represent the compressor power and turbine power per unit flow rate, respectively; s_{\text{C}} and s_{\text{T}} correspond to the entropy values at the points of adiabatic compression and adiabatic expansion, respectively; and c_{\text{p}} represents the specific heat capacity of the working fluid.

According to Eqs. (3) and (4),

(5)T_{\mathrm{L}}=T_0\frac{T_{\mathrm{T}}}{T_{\mathrm{H}}}.

For the CES system, the processes between 2‒3 involve a common temperature difference between the cooler and the heater, as well as heat leakage losses. The total process temperature difference during this process ∆T_Total is defined as

(6)\mathrm{\Delta }{T}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}= 2\mathrm{\Delta }{T}_{\mathrm{H}\mathrm{E}}+\mathrm{\Delta }{T}_{\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}}=T_{\mathrm{H}}-T_{\mathrm{T}}.

According to Eqs. (5) and (6)

(7)T_{\mathrm{L}}=T_0\left(1-\frac{\mathrm{\Delta }{T}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{T_{\mathrm{H}}}\right).

For the entire cycle 1‒2‒3‒4‒4‒5‒1 of the CES system, its cycle efficiency is defined as the energy storage efficiency, which is the ratio of expansion power to compression power. By simultaneously solving equations (1), (2), (6) and (7), we obtain:

(8)\begin{array}{rl}{\epsilon }_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}=& \frac{s_{\mathrm{T}}}{s_{\mathrm{C}}}\left(1-\frac{\mathrm{\Delta }{T}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{T_{\mathrm{H}}}\right)=\\ & \frac{s_{\mathrm{T}}}{s_{\mathrm{C}}}\left(1-\frac{2\mathrm{\Delta }{T}_{\mathrm{H}\mathrm{E}}+\mathrm{\Delta }{T}_{\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}}}{T_{\mathrm{H}}}\right),0\le \frac{s_{\mathrm{T}}}{s_{\mathrm{C}}}\le 1\end{array}

It is found that under ideal conditions, the variables that affect the efficiency of the CES cycle include the maximum discharge temperature of the compressor, the design value of the temperature difference in the heat exchanger, and the temperature difference due to the heat storage loss, which reflects the importance of temperature gradients in non-supplementary combustion gas energy storage systems. However, it should be noted that for different working fluids, the specific heat capacity and entropy during compression and expansion processes may not be exactly the same. Therefore, calculations based on ideal gases cannot fully exclude the influence of working fluid properties on system circulation efficiency.

Furthermore, irreversible losses during the compression and expansion processes must be considered, including the compressor isentropic efficiency {\epsilon }_{\text{C}}, motor mechanical efficiency ε_CM, turbine isentropic efficiency {\epsilon }_{\text{T}}, and generator efficiency {\epsilon }_{\text{TM}}. From Fig.6, it can be seen that there is a shift at points 2, 4, and 5. The deviation at 4 points is due to the increase in exhaust temperature T_H caused by irreversible compression. In order to maintain a constant temperature difference in the heat exchanger, the inlet temperature of the turbine also slightly increases.

Thus, the following equations are established

(9)\mathrm{\Delta }{T}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}= T_{\mathrm{H}}-T_{\mathrm{T}}= T_{\mathrm{H}}^{\prime }-T_{\mathrm{T}}^{\prime },

(10){\epsilon }_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}=\frac{s_{\mathrm{T}}^{\prime }}{s_{\mathrm{C}}} \left(1-\frac{2\mathrm{\Delta }{T}_{\mathrm{H}\mathrm{E}}+\mathrm{\Delta }{T}_{\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}}}{T_{\mathrm{H}}}\right){\epsilon }_{\mathrm{C}}{\epsilon }_{\mathrm{T}}{\epsilon }_{\mathrm{C}\mathrm{M}}{\epsilon }_{\mathrm{T}\mathrm{M}}.

The premise for Eq. (9) to hold is that the design values of the heat transfer temperature difference and dissipation temperature difference in the heat exchanger are identical. In practical calculations, the value of entropy is influenced by different reference values. If the influence of the total process temperature difference ∆T_Total on entropy change is ignored, setting s_{\text{T}} /s_{\text{C}} = 1 will yield the relative limit value of cycle efficiency.

(11)\begin{array}{rl}& {\epsilon }_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}^{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}=\left(1-\frac{2\mathrm{\Delta }{T}_{\mathrm{H}\mathrm{E}}+\mathrm{\Delta }{T}_{\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}}}{T_{\mathrm{H}}}\right){\epsilon }_{\mathrm{C}}{\epsilon }_{\mathrm{T}}{\epsilon }_{\mathrm{C}\mathrm{M}}{\epsilon }_{\mathrm{T}\mathrm{M}},\\ & \frac{s_{\mathrm{T}}^{\prime }}{s_{\mathrm{C}}}=1.\end{array}

In addition, it can be observed that, under a fixed pressure ratio, the number of compressor stages significantly affects the exhaust temperature at each stage, which in turn affects the heat storage temperature and cycle efficiency. The fewer compression stages, the higher the upper limit of the heat storage temperature, bringing the system closer to its ultimate cycle efficiency. Therefore, the compression level for the system proposed in Section 2 is set to 1.

Based on Eqs. (8) and (11), the cyclic efficiency of the CES system under the ideal condition, as well as the distribution of the maximum compressor discharge temperature and process temperature difference is plotted in Fig.7. At a fixed maximum compressor discharge temperature, the cyclic efficiency tends to decrease with an increase in the process temperature difference due to the increase in the system energy loss. Under constant process temperature difference, cycle efficiency increases with an increase in the maximum compressor discharge temperature, which indicates that although higher compressor discharge temperature increases the energy consumption of the compression process, the corresponding increase in the expansion process output (due to the increase of the inlet temperature) outweighs the increased compression energy demand. Therefore, increasing the compressor discharge temperature is beneficial for improving the cycle thermodynamic performance, and this benefit is especially significant when the process temperature difference is large. However, once the maximum compressor discharge temperature increases to a certain level, the increase in cycle efficiency begins levels off, and the increase in equipment cost must be considered.

Considering the equipment levels under actual engineering application conditions, the adiabatic efficiency of the compressor can be taken as 85%, the adiabatic efficiency of the turbine as 90%, and the mechanical efficiency of the motor and generator as 98%. Based on these values, the trend of the limit cycle efficiency under actual working conditions can be further obtained. As shown in Fig.8, the actual limit cycle efficiency decreases significantly compared to the ideal limit cycle efficiency shown in Fig.7. These figures establish the ideal boundary and the actual prediction boundary of the cycle efficiency of the CES system, respectively. Fig.9 further illustrates the upper limit of the cycle efficiency that can be achieved by the actual system under different configuration conditions.

The impact of variable total pressure ratio on the cycle efficiency of the system is studied. According to the Brayton thermal efficiency expression [32],

(12){\epsilon }_{\mathrm{B}\mathrm{r}\mathrm{a}\mathrm{y}\mathrm{t}\mathrm{o}\mathrm{n}}=1-{\left(\frac{1}{\pi }\right)}^{\frac{\kappa -1}{\kappa }},

where π is the total pressure ratio and κ is the polytropic index.

From the perspectives of positive and negative Brayton cycle (energy storage cycle), the difference between the expander and compressor is expressed as the external work done by the positive Brayton cycle and the work loss associated with the negative Brayton cycle, respectively. Therefore, the ratio of work loss to input work (compression work) in the inverse Brayton cycle is defined as the work loss efficiency, denoted as {\epsilon }_{\text{CWL}}. Meanwhile, it is known that the energy storage efficiency is equal to 1 −{\epsilon }_{\text{CWL}}.

(13){\epsilon }_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}}= 1-{\epsilon }_{\mathrm{C}\mathrm{W}\mathrm{L}},

(14){\epsilon }_{\mathrm{C}\mathrm{W}\mathrm{L}}=\frac{c_{\mathrm{p}}\kappa R_{\mathrm{g}}{T}_0\mathrm{\Delta }{T}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{\kappa -1}\left[\frac{1-{\left({\displaystyle \frac{1}{\pi }}\right)}^{\frac{\kappa -l}{\kappa }}}{{\pi }^{\frac{\kappa -l}{\kappa }}-1}\right],

where R_g is the ideal gas constant.

Taking the derivative of Eq. (14) can lead to , and it is found that there is an inverse correlation between work loss efficiency and pressure ratio, which implies a positive correlation between energy storage efficiency and pressure ratio. Under controlled variable conditions, a higher pressure ratio leads to a higher the energy storage efficiency. However, it is necessary to consider some objective limitations in actual applications.

4 Thermodynamic models and evaluation indicators

This section establishes corresponding thermodynamic and performance evaluation models for the research object. To highlight the influence of main parameters and reduce computational redundancy, the following reasonable assumptions are made [33,34]:

1) The proposed system operates in a stable stage before power ramp-up and ramp-down.

2) The thermal energy loss from equipment and connecting pipelines due to environmental conditions are neglected.

3) Input parameters related to equipment, such as compressors and turbines, are selected based on engineering experience.

4) CO₂ is stored under environmental conditions.

5) The system calculates the unit mass flow rate on the working fluid side, assuming equal charging and discharging times.

4.1 Thermodynamic models

4.1.1 Compressor

The compressed input electric power W_CM is

(15)W_{\mathrm{C}\mathrm{M}}= \frac{m_{\mathrm{C}}\left(h_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}\right)}{{\epsilon }_{\mathrm{C}}{\epsilon }_{\mathrm{C}\mathrm{M}}}=\frac{c_{\mathrm{p}}{T}_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}}{{\epsilon }_{\mathrm{C}}{\epsilon }_{\mathrm{C}\mathrm{M}}}\left[{\left(\frac{p_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}}{p_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}}\right)}^{\frac{\kappa -l}{\kappa }}-1\right],

where m_{\text{C}} represents the compressor mass flow rate, while h and p denote the enthalpy and pressure, respectively. The superscript “out” and “in” refer to the compressor outlet and inlet conditions.

The compressor outlet temperature is

(16)T_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}=T_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}\left\{1+\frac{1}{{\epsilon }_{\mathrm{C}}}\left[{\left(\frac{p_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}}{p_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}}\right)}^{\frac{\kappa -l}{\kappa }}-1\right]\right\}.

4.1.2 Turbine

The turbine output electric power W_TM is

(17)W_{\mathrm{T}\mathrm{M}}= \frac{m_{\mathrm{T}}\left(h_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}-h_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}\right)}{{\epsilon }_{\mathrm{T}}} {\epsilon }_{\mathrm{T}\mathrm{M}}= \frac{c_{\mathrm{p}}{T}_{\mathrm{T}}^{\mathrm{i}\mathrm{n}}}{{\epsilon }_{\mathrm{T}}} \left[{\left(\frac{p_{\mathrm{T}}^{\mathrm{o}\mathrm{u}\mathrm{t}}}{p_{\mathrm{T}}^{\mathrm{i}\mathrm{n}}}\right)}^{\frac{\kappa -l}{\kappa }}-1\right]{\epsilon }_{\mathrm{T}\mathrm{M}},

where m_T is the turbine mass flow rate.

The turbine outlet temperature is

(18)T_{\mathrm{T}}^{\mathrm{o}\mathrm{u}\mathrm{t}}= T_{\mathrm{T}}^{\mathrm{i}\mathrm{n}}\left\{1-{\epsilon }_{\mathrm{T}}\left[1-{\left(\frac{p_{\mathrm{C}}^{\mathrm{i}\mathrm{n}}}{p_{\mathrm{C}}^{\mathrm{o}\mathrm{u}\mathrm{t}}}\right)}^{\frac{1-\kappa }{\kappa }}\right]\right\}.

4.1.3 Heat exchangers

This paper uses single-shell and multi-tube shell heat exchangers arranged in reverse flow to calculate the heat exchanger efficiency. The heat exchanger efficiency ({\epsilon }_{\text{HE}}) is defined as in Fakheri [35]:

(19){\epsilon }_{\mathrm{H}\mathrm{E}}= 2{\left\{1+Z+(1+Z{)}^{1/2}\frac{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[-NTU(1+Z{)}^{1/2}\right]}{1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-NTU(1+Z{)}^{1/2}\right]}\right\}}^{-1},

where NTU represents the number of heat transfer units in the heat exchanger.

(20)NTU=\frac{L/D}{{\displaystyle \frac{0.25}{S_{t1}}}+Z{\displaystyle \frac{0.683}{S_{t2}}}} ,

where L is the length of the heat exchanger tube; D is the diameter of the heat exchanger; S_{t1} and S_{t2} are the Stanton numbers on the tube and shell sides, respectively; and Z represents the ratio of specific heat capacity of hot and cold fluids. When the substances of the cold and hot streams are unknown, Z can be taken as 1. However, when the substances of the cold and hot streams are known, the average specific heat capacity on each side is taken into account for calculation. The Stanton number is defined as

(21)S_t=\frac{0.23[1+(L/D{)}^{0.7}]}{R{e}_{}^{0.2}P{r}_{}^{0.6}},

where Re is the Reynolds number of the fluid, and Pr is the Prandtl number of the fluid.

4.2 Performance evaluation indicators

Similar to the cycle efficiency defined by Eq. (8), the cycle efficiency ε_CES of the CES system is defined as the ratio of the total power generation during the energy release process to the total energy consumption during the energy storage process, as shown in Eq. (22).

(22){\epsilon }_{\mathrm{C}\mathrm{E}\mathrm{S}}=\frac{{\displaystyle {\int }_0^{t_{\mathrm{d}\mathrm{i}\mathrm{s}}}{W}_{\mathrm{T}\mathrm{M}}\mathrm{d}t}}{{\displaystyle {\int }_0^{t_{\mathrm{c}\mathrm{h}}}{W}_{\mathrm{C}\mathrm{M}}\mathrm{d}t}},

where t represents time (s) and W represents electrical power (kW). The subscripts dis and ch represent the discharge and charging processes, respectively.

For the development of cascade heat storage in this system, the heat transfer efficiency of thermal energy can be improved as much as possible. Therefore, the heat utilization efficiency η_Thermal is defined to characterize the reheating effect of CO₂ in the system, as shown in Eqs. (23)–(25)

(23)Q_{\mathrm{C}\mathrm{S}}= t_{\mathrm{c}\mathrm{h}}{m}_{\mathrm{c}\mathrm{h}}\left[\sum _{i=1}^{n-1}\left(h_{\mathrm{C},i}^{\mathrm{i}\mathrm{n}}-h_{\mathrm{C},i}^{\mathrm{o}\mathrm{u}\mathrm{t}}\right)+\left(h_{\mathrm{C},n}^{\mathrm{i}\mathrm{n}}-h_{\mathrm{H}\mathrm{P}\mathrm{V}}\right)\right],

(24)Q_{\mathrm{T}\mathrm{S}}= t_{\mathrm{d}\mathrm{i}\mathrm{s}}{m}_{\mathrm{d}\mathrm{i}\mathrm{s}}\left[\sum _{j=2}^n\left(h_{\mathrm{T},j}^{\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{T},j}^{\mathrm{i}\mathrm{n}}\right)+\left(h_{\mathrm{T},1}^{\mathrm{i}\mathrm{n}}-h_{\mathrm{H}\mathrm{P}\mathrm{V}}\right)\right],

(25){\eta }_{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}=\frac{Q_{\mathrm{T}\mathrm{S}}}{Q_{\mathrm{C}\mathrm{S}}},

where Q represents heat exchange (kW), m represents the mass flow rate (kg/s), and n represents the number of heat exchanger stages in the heat storage system. h represents the enthalpy value of the working fluid (kJ/kg), and the subscripts CS and TS represent the heat storage process and the reheating process, respectively.

5 Results and discussion

5.1 Model validation

This section validates the previously described modeling methods. Since the proposed CES is entirely new, the relevant parameters in Zhang et al. [36] were compared with the predicted values. As shown in Tab.3, the maximum error in pressure is 1.4%, and the maximum error in temperature is 1.0%. Therefore, the model developed in this work can be used for further performance analysis, with the differences in calculations being negligible.

5.2 Typical design conditions and result analysis

According to the CES system proposed in this paper, the specific parameter settings [25–28] are shown in Tab.4.

Further, the calculation results of the main performance parameters of the CES system (shown in Tab.5) and the thermodynamic parameters of each node (shown in Tab.6) are obtained. Research has found that under typical design conditions, the output power per unit mass flow rate is 334 kW/(kg·s^‒1), the cycle efficiency of the CES system reaches 76.4%, and the thermal utilization efficiency reaches 95.9%. Additionally, suing Eq. (11), the theoretical ultimate efficiency of the system under rated input conditions was calculated to be 73.5%, which differs from the actual calculated value by 3.95%. This further verifies the good applicability of the theoretical algorithm proposed in this paper, with the difference between the two values attributed to the difference in calculation methods and the influence of physical properties.

Next, the graded heat storage component of this system is further analyzed, which consists of 4 pairs of cooling/heating heat exchangers and 4 sets of heat storage units. These heat storage units are sequentially connected in series from the high-temperature zone to the low temperature zone, using molten salt, thermal oil, pressurized water, and atmospheric water as the medium. Fig.10 and Fig.11 illustrate the heat transfer process of the graded heat storage unit during the energy storage and release processes. It is evident that, although the heat transfer curve becomes curved due to a sudden change in specific heat at constant pressure near the critical temperature range of CO₂, both the hot-side flow and the cold-side flow exhibit a good match. The ultimate positive result is an increase in the overall heat storage temperature, an increase in heat utilization efficiency, an increase in the turbine inlet temperature, a reduction in unnecessary heat loss, and, consequently, a maximum increase in power generation.

Additionally, from the temperature difference change curve, it can be found that the trend of the heat transfer temperature difference between the first-stage heat storage unit during energy storage and the first-stage heat storage unit during energy release is opposite. This is due to the different target values set for the heat exchanger under the two working conditions (the cooling temperature of CO₂ during energy storage and the heating temperature of CO₂ during energy release). Furthermore, extreme temperature difference points (indicating high-efficiency or low-efficiency heat transfer) were found in the middle of the heat transfer process in the second, third, and fourth heat storage units. At these extreme points, the corresponding temperatures of the cold and hot flows were essentially the same. It is speculated that this phenomenon may be attributed to changes in the density, specific heat, and other physical properties of the corresponding heat transfer material at these locations, which resembles the “pinch point” effect commonly encountered in heat exchanger design.

5.3 Performance analysis

The previous analysis has highlighted the advantages of the new CES system. This section will further investigate the impact of key parameters on system performance under off-design conditions. Based on the previous discussion, the important parameters selected for analysis include the total pressure ratio, total expansion ratio, minimum heat transfer temperature difference, heat exchanger efficiency, and ambient temperature.

5.3.1 Total pressure ratio (π_C) and total expansion ratio (π_T)

As discussed in Section 3, the total pressure ratio is positively correlated with cycle efficiency. In this study, the total pressure ratio was adjusted by modifying the outlet pressure of the compressor, which in turn affected both the high-pressure CO₂ pressure (p_{\text{HPV}}) and the inlet pressure (p_{\text{T}}) of the turbine. Due to pressure drops in the heat exchanger and the outlet stabilizing pressure drop on the high-pressure side of the CES system, the total expansion ratio increases or decreases with changes in total pressure ratio (as the expansion ratio is a decimal, it is represented as the reciprocal in Fig.12). At {\pi }_{\text{C}} = 66.67, a turning point in the maximum heat storage temperature is observed, mainly due to the influence of CO₂ enthalpy changes on the outlet temperature of the compressor at that point.

From Fig.13, it can be observed that the theoretical ultimate efficiency shows a gradual increase within the compressed outlet pressure range of 5.6 to 9.2 MPa, validating the mathematical expression in Section 3 and the actual growth in cycle efficiency. Notably, the cycle efficiency increases significantly between the compression outlet pressure from 6.4 and 6.8 MPa, reaching its peak at 6.8 MPa. This is because the pressure is close to the critical pressure, which increases the enthalpy difference between the inlet and outlet of the turbine, thus significantly boosting power generation. This suggests that while the total pressure ratio has a positive impact on cycle efficiency, there are also specific “optimization points” near the critical pressure range. These points may help designers strike a balance between system performance and investment costs.

5.3.2 Heat exchange temperature difference

The temperature difference in heat exchange plays a significant role in determining the storage and reheating temperatures of the CES system, as shown in Fig.14. In this study, the default heat storage refrigerant flow rate, CO₂ flow rate, and the set temperature at the CO₂ outlet of the cooler remain unchanged. It is observed that both T_1st and T_T decrease with the increase in heat transfer temperature difference, with T_T showing a more pronounced reduction due to the influence of two heat exchanges. On the other hand, T_2nd, T_3rd, and T_4th are less affected by the heat transfer temperature difference. This is mainly because the cooling load obtained by the internal heat exchanger becomes more substantial as temperature difference increases. Therefore, an increase in the heat transfer temperature difference leads to a decrease in heat utilization efficiency.

According to Fig.15, it can be further observed that as the temperature difference in heat exchange increases, both the system cycle efficiency and thermal utilization efficiency experience a significant decrease. The downward trend of both efficiencies is basically consistent, demonstrating the thermoelectric coupling relationship between “compression process (electricity → heat)” and “expansion process (heat → electricity).”

5.3.3 Heat exchanger efficiency

In previous studies, it was often assumed that the heat exchanger efficiency remains constant and equal across different conditions. However, based on the thermodynamic model of the heat exchanger presented in Section 3, it becomes evident that the heat exchanger efficiency is influenced by factors such as fluid flow rate, the thermal properties of both cold and hot fluids, operating conditions, and the structure of the heat exchanger. As shown in Fig.16, within the growth range of heat exchanger efficiency {\epsilon }_{\text{HE}} from 0.70 to 1.00, both {\epsilon }_{\text{CES}} and {\eta }_{\text{Thermal}} gradually increase. Specifically, for every 0.5 increase in {\epsilon }_{\text{HE}}, {\epsilon }_{\text{CES}} increases by approximately 2.3%, while η_Thermal increases by around 2.7%, fully demonstrating the significant impact that heat exchanger efficiency has on system performance. In practical applications, heat exchanger efficiency often needs to be at least 0.90 or above to be considered reasonable, and it plays a critical role in determining the heat storage temperature (T_T) in the CES system.

5.3.4 Ambient temperature

For CES systems operating in different regions or under varying seasonal conditions, environmental temperature plays a significant role in their performance. As shown in Fig.17, within the ambient temperature range of 274‒308 K, both the system compression power (W_C) and turbine power generation (W_T) increase as the ambient temperature rises. The theoretical limit efficiency (ε_limit) increases slightly from 73.07% to 73.34%, while the cycle efficiency ({\epsilon }_{\text{CES}}) decreases from 77.03% to 75.80%. This behavior occurs because, although the increase in heat exchange temperature increases the inlet and outlet temperatures of the compressor, theoretically improving the heat storage temperature and theoretical maximum efficiency, the increase in compression work is significantly higher than that in turbine power generation. From a practical perspective, a higher compressor inlet temperature generally results in a lower isentropic efficiency. Therefore, when using {\epsilon }_{\text{limit}} as the criterion for environmental temperature influence, it is necessary to make corrections based on the actual situation.

5.4 System comparison and innovation

To highlight the innovation and benefits of the CES system proposed in this paper, a comparison was made with similar systems proposed by Wu et al. [37] and Zhang et al. [38]. Detailed descriptions of these two systems can be found in the Appendix. As shown in Tab.7, the CES system presented in this paper achieves a cycle efficiency of up to 76.40% without significantly increasing the high-pressure. This demonstrates that the CES system is both technologically innovative and economically viable. However, it is worth noting that due to the limited volume of low-pressure gas storage containers, the energy storage density of the system is relatively low, which means land usage requirements need to be carefully considered when applying in engineering practice.

6 Conclusions

This paper proposes a CES system featuring a high-temperature graded heat storage structure, establishes guiding thermodynamic intrinsic cycles, and provides performance evaluation standards. Additionally, it conducts sensitivity analysis of off-design key parameters based on typical design systems. The following conclusions are drawn:

1) Both a basic and a completely irreversible thermodynamic intrinsic cycle were established for the CES system based on the forward/reverse Brayton cycle. A theoretical limit efficiency and work loss efficiency calculation method was proposed integrating finite scale analysis method of heat transfer. It was also found that the maximum discharge temperature of the compressor, the design temperature difference in the heat exchanger, and the temperature difference in heat storage losses are the most significant factors affecting the theoretical limit efficiency. Within a certain range, work loss efficiency inversely correlates with the pressure ratio. This approach provides a novel framework for optimizing the design of CES or similar compressed gas energy storage systems.

2) The proposed high-temperature graded heat storage structure significantly enhances the heat transfer efficiency and overall system performance across a wide temperature range. Under typical design conditions, the cycle efficiency of the CES system reaches 76.4%, the thermal utilization efficiency reaches 95.9%, and the unit mass flow rate work index is 334 kW/(kg·s^‒1). The theoretical limit efficiency of 73.5% differs by 3.95% from the actual calculated value, further validating the applicability the applicability of the proposed algorithm.

3) Under constant conditions of graded thermal storage unit parameters and compressor/turbine isentropic efficiency, the larger the total pressure ratio, the smaller the minimum heat transfer temperature difference, leading to a higher the system cycle efficiency. Moreover, reducing the heat transfer temperature difference significantly boosts heat utilization efficiency. The impact of heat exchanger efficiency on system performance aligns with the heat transfer temperature difference, though heat exchanger efficiency has a more significant impact. The theoretical efficiency limit due to environmental temperature contrasts with the actual cycle efficiency change trend, primarily because the theoretical calculation assumption neglects the influence of the physical properties of working fluid and the performance of power equipment under different operating conditions. This discrepancy should be considered when conducting confirmatory analysis.

It is worth noting that the results of this study still require further experimental validation, especially concerning the performance of the new high-temperature and high-load compressors and turbines. The performance of key equipment is an important guarantee for the theoretical calculation of cycle efficiency.

Notations

Appendix

Construction of CES system in Wu et al. [37] and Zhang et al. [38]

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