Analysis of heat transfer characteristics of a novel liquid CO₂ energy storage system basedon two-stage cold and heat storage

Abstract

As the installed capacity of renewable energy such as wind and solar power continues to increase, energy storage technology is becoming increasingly crucial. It could effectively balance power demand and supply, enhance allocation flexibility, and improve power quality. Among various energy storage technologies, liquid CO₂ energy storage (LCES) stands out as one of the most promising options due to its advantages such as high round-trip efficiency (RTE), high energy storage density (ESD), safety, stability, and longevity. Within the system, the cold and heat storage units play a critical role in determining the overall performance of the system and are particularly important among its various components. In this paper, a novel LCES system is proposed and the heat transfer characteristics are analyzed in detail. Then, the impact of key parameters on the liquefaction ratio and RTE is discussed. The results indicate that the RTE, ESD, and exergy efficiency of the system are 56.12%, 29.46 kWh/m³, and 93.73% under specified design conditions, respectively. During the gas–liquid phase change process of carbon dioxide or when it is in a supercritical state, the related heat transfer processes become more complex, leading to increased energy loss. The analysis of key parameters of the Linde-Hampson liquefaction unit reveals that as the liquefaction temperature decreases, both the liquefaction ratio and RTE increase. While the liquefaction pressure has a minimal impact on the liquefaction ratio, it significantly affects RTE, with an optimal liquefaction pressure identified.

Graphical abstract

Keywords

liquid CO₂ energy storage / graded cold and heat storage / heat transfer characteristics / liquefaction ratio / thermodynamic analysis

1 Introduction

Under the guidance of new energy system construction, renewable energy is becoming the main source of electricity. Considering the volatility, intermittence, and unpredictability of renewable energy power generation, the large-scale and long-time energy storage technology is playing an increasingly important role. CO₂ energy storage (CES) has attracted much attention in recent years because of its special advantages such as high round-trip efficiency (RTE) and security, long life, and strong flexibility [1]. Several mature CES projects had been completed in the world by the end of 2023. However, the common technical approaches primarily utilize a gas-liquid two-phase state for the working medium storage. The large volume of low-pressure gaseous CO₂ makes the energy storage density (ESD) of the system too small and limits the wide application. Therefore, liquid CES (LCES) was proposed to improve the system compactness and reduce the land occupation [2]. An analysis of the extensive published literature indicates that the research hotspots of the LCES system mainly focus on analysis and optimization of the whole system, coupling characteristics with external energy sources, and design of key units and components.

From the perspective of the system, both thermodynamic and economic evaluations, as well as dynamic operating characteristics and off-design behavior, are important areas for research. Zheng et al. [3] conducted a comprehensive thermodynamic and economic analysis of four CES systems based on different working medium storage modes and found that the system whose liquid CO₂ was stored in the high-pressure and low-pressure tanks was the most competitive. Ma and Liu [4] suggested an LCES system with low-pressure storage. The thermodynamic and economic evaluation results indicated that the maximum RTE, peak ESD, and lowest levelized cost were 56.23%, 19.90 kWh/m³, and 0.0907$/kWh, respectively. Wan et al. [5] developed an off-design performance prediction model based on preliminary designs of turbomachinery components and heat exchangers to evaluate the feasibility of the LCES system and proposed constant pressure discharging and sliding pressure discharging operation strategies to cope with the load fluctuations.

New energy sources such as wind, solar and geothermal energy, along with conventional thermal power plants and liquid natural gas (LNG), are often integrated with the LCES system to achieve improved overall performance. Fu et al. [6] proposed a new LCES technology coupled with wind farm and PV panels, and investigated the optimal scheduling strategy in four different application scenarios. This design could significantly reduce the electricity purchasing investment and improve the scheduling flexibility. Chae and Lee [7] introduced an integration concept of the LCES system with the steam cycle of the thermal power plant. The optimization analysis achieved the maximum RTE of 46% and ESD of 36 kWh/m³. Zhang et al. [8] showed a combined cooling, heating, and power system based on liquid CO₂ energy storage, wind power and wind turbine waste heat utilization. The system power efficiency, system energy efficiency, and exergy density were 0.48, 1.19, and 11.51 kWh/m³, respectively.

A typical LCES system consists of cold/heat storage units, a working medium storage unit, and compression/expansion units. The cold and heat storage units are particularly significant, as they have a substantial impact on system efficiency. To enhance the efficiency of these storage units, the use of high-performance materials, high-efficient heat exchangers, and cascaded heat transfer technology are considered effective approaches. Hüttermann et al. [9] emphasized the significant influence of the temperature-dependence of the heat capacity on the performance of the thermal energy storage system and investigated 9 real and further hypothetical storage materials. A general formulation in terms of summarizing key figures was developed to guide the material selection. Liu et al. [10] completed the thermal and hydraulic design of a recuperator in the LCES system. In addition, the practicability of the heat exchanger was proved by the analysis of the temperature and heat transfer capacity of CO₂ at different loads. Zhao et al. [11] used a discretization method to design heat exchangers considering the real physical properties of CO₂. The optimal RTE value could reach 65.16%. Fu et al. [12] established a CO₂ mixtures energy storage system with two-stage cold storage processes. The calcium chloride solution and ice slurry were used in conjunction to realize the liquefaction and evaporation of low-pressure CO₂. The results showed that under the design conditions, the efficiency and energy density were respectively 60.12% and 14.19 kWh/m³.

However, most studies focus primarily on macroscopic process or structure design, leaving a deficiency in the detailed analysis of intrinsic heat transfer characteristics. Additionally, the effect of key parameters on the system performance remains unclear. To address this issue, a novel LCES system incorporating the Linde-Hampson (L-H) liquefaction cycle and two-stage cold/heat storage processes is established in this paper. The various storage methods and materials are compared and selected. Then, the heat transfer characteristics are analyzed in detail. Finally, the influence of key parameters on system performance along with the underlying mechanism is explored. This paper makes a significant scientific contribution to the current literature by establishing a novel LCES system that incorporates the L-H liquefaction cycle and two-stage cold/heat storage processes, enabling low-pressure liquid phase storage of CO₂ on both sides. Additionally, it elucidates the heat transfer characteristics through detailed T–Q curves and exergy analysis, leading to valuable design principles heat exchangers. Furthermore, it thoroughly investigates the influence of key parameters on the L-H liquefaction cycle, as well as the underlying thermodynamic mechanism.

2 Cold/heat storage method and media

2.1 Storage method

Cold/heat storage methods include sensible heat storage, latent heat storage, and thermo-chemical heat storage. In the LCES system, common storage methods include the two-tank liquid media system, the direct or indirect contact packed bed solid media system, and the phase change material (PCM) storage system. Liquid phase storage has the advantages of uniform temperature distribution and flexible control of output cooling/heating capacity [13]. Additionally, the structure is easier to design and construct. Commonly used materials include propane and methanol for cold storage, as well as water and thermal oil for heat storage. These materials serve dual purposes as both heat transfer fluid and storage medium. Solid phase storage typically utilizes materials such as pebbles, basalts, and granite, which have stable chemical properties and provide higher safety, enabling a wider operating temperature range. However, the accompanying severe thermocline phenomenon can significantly hinder system performance. In addition, these materials generally have low specific heat capacities, necessitating a larger quantity of material. In contrast, phase change storage leverages the latent heat of materials during phase changes, offering a more efficient storage solution. A significant advantage of this method is its ability to mitigate the effects of thermocline [14]. Common PCMs include inorganic salts, organic substances, and composite media. However, there has yet to be any practical application of an LCES system based on phase change storage. With advancements in technology and materials, it is likely that such applications will become more feasible in the near future.

2.2 Storage medium

Taking these reasons into consideration, this paper selects the two-tank liquid media storage strategy. Considering that the cold storage temperature ranges from approximately −50 to −20 °C, the main thermophysical properties of potential cold storage media are shown in Fig.1. Some authoritative public references [15,16] and National Institute of Standards and Technology （NIST） REFPROP provide data sources. Although it is evident that propane has a significantly lower viscosity than the others, which contributes to reducing the flow resistance, it could not significantly reduce the power consumption of the pump according to engineering experience. However, methanol has the benefits of high density, heat conductivity, and specific heat as well as low cost, which is conducive to improving the thermal energy efficiency of the system while reducing the cost and occupied land. Therefore, methanol is the best choice under comprehensive consideration. Tab.1 lists the common high-temperature heat storage materials [17,18]. In view of the operation temperature and large specific heat capacity, thermal oil and pressurized water are selected to establish a two-stage heat storage process for improving heat utilization efficiency, though thermal oil does not have an advantage in price.

3 System modeling and analysis

3.1 System description

A novel LCES system based on cascaded cold and heat storage and the L-H liquefaction cycle is established, whose schematic diagram is demonstrated in Fig.2. To decrease the difficulty in heat exchanger design, reduce the cooling loss, and improve the cold storage efficiency, the cold storage unit is configured as two stages: sensible heat and latent heat exchange processes. Moreover, considering the excessive temperature range of the heat storage, the two-stage heat storage process of thermal oil and water is also designed to achieve the energy cascade utilization. The operation principle of the whole system includes two processes: charging and discharging.

In the charging process, the low-pressure and low-temperature liquid CO₂ released from the CO₂ tank1 (CT1) releases cooling after being regulated by the throttle1 (TV1) (1–2) through a two-stage cold storage process (2–4) and turns into gas. Methanol is used as cold storage medium to absorb latent and sensible cold energy (22–23, 26–27) through methanol heat1 (MH1) and methanol heat2 (MH2), respectively. Then, the CO₂ is compressed to supercritical phase in the compressor (5–6) and chilled down. The compression heat is absorbed by a two-stage heat transfer process in the oil cooler (OC) (6–7) and the water cooler (WC) (7–8), where the thermal oil and pressurized water from the oil tank2 (OT2) and water tank2 (WT2) are heated (30–31, 34–35), respectively. The resulting hot media are stored in the oil tank1 (OT1) and water tank1 (WT1). Next, the CO₂ flows into a multi-flow heat exchanger (MHE) to be cooled to liquid phase (8–9) and then experiences a large pressure drop through the throttle2 (TV2) (9–10). After that, the gas-liquid mixture enters a separator (SEP), where the CO₂ gas returns to MHE to release sensible heat (11–12) and is compressed again while the liquid CO₂ is stored in the CT2 (10–13). Much low-temperature methanol is required in MHE where the thermal energy of CO₂ is absorbed. Finally, the methanol heated flows into the high-temperature methanol tank (MT5) (38–39). In this case, the electric energy is converted into potential energy and thermal energy.

In the discharging process, the CO₂ released from the CT2 expands through the turbine (18–19) after being heated by the methanol heater3 (MH3) (15–16), water heater (WH) (16–17) and the oil heater (OH) (17–18) successively. The turbine drives the generator to produce electricity. The gas CO₂ at the exit of the turbine absorbs cooling from the cold storage medium and is cooled to low-temperature liquid phase (19–21) and then stored in CT1 for the next cycle. The methanol cooler1 (MC1) and methabol cooler2 (MC2) are key componets to achieve the liquefaction of CO₂. The potential energy and thermal energy are reconverted into electricity.

3.2 Mathematical modeling

The Aspen plus software is used to simulate the operation of the system. Before modeling, it is assumed that [19,20] the pressure drop of carbon dioxide in each heat exchanger is constant; the heat dissipation of the equipment and pipes as well as pressure drop in pipes is negligible; the system is subjected to steady-state condition during the simulation analysis; the efficiency of the motor and generator are assumed as 100%, and the running time of the charge process and the discharge process is the same.

The conservation of energy and mass as well as the entropy production expressed in Eqs. (1)–(3) are the basic principles of operation [21].

(1)\sum _{}^{} {\dot{m}}_{\mathrm{i}\mathrm{n}}= \sum _{}^{} \dot{m}{}_{\mathrm{o}\mathrm{u}\mathrm{t}},

(2){\dot{Q}}_{\mathrm{i}\mathrm{n}}+ \sum _{}^{} \left(\dot{mh}\right){}_{\mathrm{i}\mathrm{n}} +\dot{W}{}_{\mathrm{i}\mathrm{n}}={\dot{Q}}_{\mathrm{o}\mathrm{u}\mathrm{t}}+\sum _{}^{}\left(\dot{mh}\right) {}_{\mathrm{o}\mathrm{u}\mathrm{t}}+{\dot{W}}_{\mathrm{o}\mathrm{u}\mathrm{t}},

(3)\sum \frac{\dot{Q}}{T}+\dot{{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{n}}}=\sum {\dot{m}}_{\mathrm{o}\mathrm{u}\mathrm{t}}{s}_{\mathrm{o}\mathrm{u}\mathrm{t}}-\sum {\dot{m}}_{\mathrm{i}\mathrm{n}}{s}_{\mathrm{i}\mathrm{n}},

where \dot{m}, \dot{Q}, h, s, and \dot{W} denote the mass flow rate, the heat transfer rate, specific enthalpy, specific entropy, and mechanical power, in sequence; \dot{{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{n}}} is the entropy generation, and the subscript in and out indicate the inflow and outflow of the system.

3.2.1 Compressor model

The compressor isentropic efficiency formula is defined in [22,23],

(4){\eta }_{\mathrm{c}}= \frac{h_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}\mathrm{s}}-h_{\mathrm{i}\mathrm{n}}}{h_{\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{i}\mathrm{n}}},

in which {\eta }_{\mathrm{c}} represents compressor isentropic efficiency, and the subscript “is” indicates the isentropic progress.

The power consumption of each compressor can be calculated by

(5)W_{\mathrm{c},i}=\frac{\kappa }{\kappa -1}\frac{m_{\mathrm{c}}{R}_{\mathrm{c}}{T}_{\mathrm{c},i}^{\mathrm{i}\mathrm{n}}}{{\eta }_{\mathrm{c},i}}\left[{{\beta }_{\mathrm{c},i}}^{\frac{\kappa -1}{\kappa }}-1\right],

in which i represents the stage of the compressor, \kappa is the ratio of specific heat, {\beta }_{\mathrm{c},i} is the compression ratio of each compressor, R_{\mathrm{c}} is the gas constant of CO₂, and T_{\mathrm{c},i}^{\mathrm{i}\mathrm{n}} is the CO₂ temperature at the compressor inlet.

The compressor outlet temperature is calculated by

(6)T_{\mathrm{c},i}^{\mathrm{o}\mathrm{u}\mathrm{t}}= \frac{T_{\mathrm{c},i}^{\mathrm{i}\mathrm{n}}\left({\beta }_{\mathrm{c},i}^{\frac{\kappa -1}{\kappa }}-1\right)}{{\eta }_{\mathrm{c},i}}+T_{\mathrm{c},i}^{\mathrm{i}\mathrm{n}}.

3.2.2 Turbine model

Similar to the compressor, the turbine isentropic efficiency formula is

(7){\eta }_{\mathrm{t}}= \frac{h_{\mathrm{i}\mathrm{n}}-h_{\mathrm{o}\mathrm{u}\mathrm{t}}}{h_{\mathrm{i}\mathrm{n}}-h_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}\mathrm{s}}}.

The power generated from each expander stage is

(8)W_{\mathrm{t},i}=\frac{\kappa }{\kappa -1}{m}_{\mathrm{t}}{R}_{\mathrm{c}}{T}_{\mathrm{t},i}^{\mathrm{i}\mathrm{n}}\left[1-{{\beta }_{\mathrm{t},i}}^{-\frac{\kappa -1}{\kappa }}\right],

in which i stands for the stage of the turbine, {\beta }_{\mathrm{t},i} is the expansion ratio of each turbine, and T_{\mathrm{t},i}^{\mathrm{i}\mathrm{n}} is the CO₂ temperature at the turbine inlet.

The turbine outlet temperature is calculated by

(9)T_{\mathrm{t},i}^{\mathrm{o}\mathrm{u}\mathrm{t}}=T_{\mathrm{t},i}^{\mathrm{i}\mathrm{n}}-T_{\mathrm{t},i}^{\mathrm{i}\mathrm{n}}{\eta }_{\mathrm{t},i}\left(1-{\beta }_{\mathrm{t},i}^{\frac{\kappa -1}{\kappa }}\right).

3.2.3 Heat exchanger model

Heat exchangers are important components in the energy storage system, which determine the heat energy utilization efficiency and overall system performance. Enhanced logarithmic mean temperature difference method is used in the design process. Based on the design principle, there are

(10)Q_{\mathrm{h}\mathrm{e}}=m_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}}(h_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d},\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d},\mathrm{i}\mathrm{n}})=m_{\mathrm{h}\mathrm{o}\mathrm{t}}(h_{\mathrm{h}\mathrm{o}\mathrm{t},\mathrm{i}\mathrm{n}}- h_{\mathrm{h}\mathrm{o}\mathrm{t},\mathrm{o}\mathrm{u}\mathrm{t}}),

(11)Q_{max}=min(m{}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}} (h{}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d},\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}-h_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d},\mathrm{i}\mathrm{n}} ),m{}_{\mathrm{h}\mathrm{o}\mathrm{t}}(h_{\mathrm{h}\mathrm{o}\mathrm{t},\mathrm{i}\mathrm{n}}-h_{\mathrm{h}\mathrm{o}\mathrm{t},\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}})),

(12){\epsilon }_{\mathrm{h}\mathrm{e}}=\frac{Q_{\mathrm{h}\mathrm{e}}}{Q_{\mathrm{m}\mathrm{a}\mathrm{x}}},

in which Q_{\mathrm{h}\mathrm{e}} is the heat flux, Q_{\mathrm{m}\mathrm{a}\mathrm{x}} indicates the theoretical maximum heat load, and h_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d},\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}} indicates the specific enthalpy of the cold fluid when its outlet temperature is equal to the hot fluid inlet temperature. Similarly, h_{\mathrm{h}\mathrm{o}\mathrm{t},\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}} indicates the specific enthalpy of the hot fluid when its outlet temperature is equal to the cold fluid inlet temperature. It represents the limiting case during the heat exchange progress. {\epsilon }_{\mathrm{h}\mathrm{e}} indicates the heat exchanger effectiveness.

3.2.4 Throttle model

The enthalpy of the fluids at the inlet and outlet are considered as the same, expressed as

(13)h_{\mathrm{i}\mathrm{n}}=h_{\mathrm{o}\mathrm{u}\mathrm{t}}.

3.2.5 Pump model

The pump isentropic efficiency is determined by

(14){\eta }_{\mathrm{p}}= \frac{h_{\mathrm{o}\mathrm{u}\mathrm{t},\mathrm{i}\mathrm{s}}-h_{\mathrm{i}\mathrm{n}}}{h_{\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{i}\mathrm{n}}}.

The work required by the pump is calculated by

(15)W_{\mathrm{p}}= m_{\mathrm{c}{\mathrm{o}}_2}(h_{\mathrm{o}\mathrm{u}\mathrm{t}}-h_{\mathrm{i}\mathrm{n}}).

3.2.6 Liquid storage tank model

Artificial liquid storage tanks are used to store liquid CO₂ at different pressures and temperatures and good thermal insulation performance could be guaranteed. The processes of streams flowing through CT1 and CT2 are taken as isothermal and isobaric. Thus, at the inlet and outlet of CT1 and CT2, there are

(16)T_{\mathrm{i}\mathrm{n}}=T_{\mathrm{o}\mathrm{u}\mathrm{t}},

(17)P_{\mathrm{i}\mathrm{n}}=P_{\mathrm{o}\mathrm{u}\mathrm{t}},

(18)V_{\mathrm{C}\mathrm{T}1} =\frac{m_{\mathrm{C}{\mathrm{O}}_2}\cdot t_{\mathrm{c}}}{{\rho }_{\mathrm{C}{\mathrm{O}}_2,1}},

(19)V_{\mathrm{C}\mathrm{T}2} =\frac{m_{\mathrm{C}{\mathrm{O}}_2}\cdot t_{\mathrm{d}}}{{\rho }_{\mathrm{C}{\mathrm{O}}_2,2}},

where V_{\mathrm{C}\mathrm{T}1} and V_{\mathrm{C}\mathrm{T}2} mean the volume of CT1 and CT2 respectively, and {\rho }_{\mathrm{C}{\mathrm{O}}_2,1} is the density of CO₂ stored in the CT1 while {\rho }_{\mathrm{C}{\mathrm{O}}_2,2} is the density of CO₂ stored in the CT2.

3.2.7 Evaluation indicator

(1) Round-trip efficiency

RTE is the ratio of total electricity generation to total electricity consumption in a complete charge-discharge cycle, which is described by [24,25]

(20)\mathrm{R}\mathrm{T}\mathrm{E}=\frac{{\int }_0^{t_{\mathrm{d}}}{W}_{\mathrm{T}}\left(t\right)\mathrm{d}t}{{\int }_0^{t_{\mathrm{c}}}{W}_{\mathrm{C}}\left(t\right)\mathrm{d}t+{\int }_0^{t_{\mathrm{d}}}{W}_{\mathrm{P}}\left(t\right)\mathrm{d}t} ,

where W_{\mathrm{T}} is the turbine power output, W_{\mathrm{C}} is the compressor power input, W_{\mathrm{P}} is the pump power input, t_{\mathrm{c}} is energy charge time, and t_{\mathrm{d}} is energy discharge time.

(2) Energy storage density

ESD equals the ratio of total power generation to total volume of working medium, which is expressed as [5,11,26]

(21)\mathrm{E}\mathrm{S}\mathrm{D}=\frac{W_{\mathrm{T}}\cdot t_{\mathrm{d}}}{V_{\mathrm{C}\mathrm{T}1}+V_{\mathrm{C}\mathrm{T}2}},

where V_{\mathrm{C}\mathrm{T}1} and V_{\mathrm{C}\mathrm{T}2} denote the volume of the CT1 and the CT2, respectively.

3.3 Exergy analysis model

From the viewpoint of the Second Law of Thermodynamics, exergy is denoted as the ability of the maximum useful work done to the objective environment in any variable process. It also reflects the amount of energy loss and the potential for improvement.

The exergy balance equation of each equipment is shown in Eq. (22).

(22)\dot{E_{\mathrm{F},k}}= \dot{E_{\mathrm{P},k}}+ \dot{E_{\mathrm{D},k}},

where \dot{E_{\mathrm{F}}} is the fuel exergy, i.e., the input exergy; \dot{E_{\mathrm{P}}} is the output exergy as a product; \dot{E_{\mathrm{D}}} is the exergy destruction owing to irreversibility or some technical limitations; the subscript k represents the object; and the fuel exergy and product exergy for heat exchangers are defined in Tab.2.

To better evaluate the performance of equipment, the exergy efficiency {\eta }_{\mathrm{e}\mathrm{x},k} and the relative exergy destruction y_{\mathrm{D},k}^{\ast } are respectively defined in Eqs. (23) and (24).

(23){\eta }_{\mathrm{e}\mathrm{x},k}=\frac{\dot{E_{\mathrm{P},k}}}{\dot{E_{\mathrm{F},k}}},

(24)y_{\mathrm{D},k}^{\ast }=\frac{\dot{E_{\mathrm{D},k}}}{\dot{E_{\mathrm{D},\mathrm{t}\mathrm{o}\mathrm{t}}}}.

The system exergy balance equation is expressed as [23]

(25)\dot{E_{\mathrm{F},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}=\dot{E_{\mathrm{P},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}+\dot{\sum E_{\mathrm{D},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}+\dot{E_{\mathrm{L},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}},

where {\dot{E}}_{\mathrm{F},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} and {\dot{E}}_{\mathrm{P},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} refer to the fuel exergy input and product exergy output of the whole system, {\dot{E}}_{\mathrm{D},\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} is the exergy destruction, and {\dot{E}}_{\mathrm{L}} is the exergy loss to the environment, which is independent of specific components.

The system exergy efficiency {\eta }_{\mathrm{e}\mathrm{x}} is the ratio of the amount of total exergy output to total exergy input, which can be defined as

(26){\eta }_{\mathrm{e}\mathrm{x}}=\frac{{\int }_0^{t_{\mathrm{d}}}{W}_{\mathrm{T}}\mathrm{d}t+\sum {\dot{m}}_{\mathrm{o}\mathrm{u}\mathrm{t}}{e}_{\mathrm{o}\mathrm{u}\mathrm{t}}}{{\int }_0^{t_{\mathrm{c}}}{W}_{\mathrm{C}}\mathrm{d}t+{\int }_0^{{\mathrm{t}}_{\mathrm{d}}}{W}_{\mathrm{P}}\mathrm{d}t+\sum {\dot{m}}_{\mathrm{i}\mathrm{n}}{e}_{\mathrm{i}\mathrm{n}}}.

3.4 Model validation

To improve the accuracy of performance analysis, the model of the system proposed has been validated by comparing the results with the published data. The RTE and ESD are selected to verify the reliability of models. As observed in Tab.3, the relative deviations are both within 5%. Therefore, the models built in this paper are acceptable for further performance analysis as the differences are reasonably negligible.

4 Results and discussion

4.1 Designed working conditions

Thermophysical properties of substances are acquired from the REFPROP database and the classical P–R equation is selected. Some main design parameters are summarized in Tab.4 according to the previous research [7,28,29].

Under the design condition, the RTE, ESD, and exergy efficiency of the system are 56.12%, 29.46 kWh/m³, and 93.73%, respectively. Of the many physical properties of CO₂, specific heat is the most important factor affecting the heat transfer process. It is recognized that the specific heat of CO₂ changes with different states, especially near the critical point (7.38 MPa, 31 °C), as depicted in Fig.3.

4.2 Heat transfer characteristics analysis

To better clarify the heat transfer characteristics, exploit the energy saving potential, and obtain further optimization direction, the composite T–Q curves for the system are illustrated in Fig.4. The cold storage unit is composed of sensible heat and latent heat storage processes. In MH1 and MC1, CO₂ undergoes the gas-liquid phase change while there are only single-phase flow and heat transfer phenomenon in the MH2 and MC2.

Considering the complexity of heat transfer processes, it is necessary to further explain the comment in Fig.2, where c and d mean the charging process and discharging process, respectively. Arabic numerals represent the different heat transfer stages of the specific medium in the system. Therefore, the temperature of CO₂ remains almost constant in the first-stage heat exchange process while the temperature of methanol is constantly changing. The mismatch leads to a relatively large exergy destruction. The heat transfer temperature difference is in the range of 3 to 10 °C. In the second-stage heat exchange process, the temperatures of CO₂ and methanol are both changing. The alignment between the heat transfer curves shows a higher degree of satisfaction. The heat transfer temperature difference is in the range of 4 to 7 °C.

A two-stage heat storage process combining thermal oil and pressurized water is employed. The heat storage temperature ranges of the two stages are approximately 200–340 °C and 20–170 °C, respectively. In the first-stage heat storage process, the heat transfer temperature difference in the discharging process gradually increases and reaches the maximum value of approximately 45 °C. In addition, during the charging process, the temperature difference first decreases and then rises slightly. This is primarily due to the decrease in the specific heat capacity of thermal oil as the temperature increases. On the other hand, the specific heat of CO₂ changes only slightly at high temperature and pressure. But in the second-stage, the specific heat of CO₂ first increases and then decreases with the increase of the temperature. Moreover, a peak value is observed when the temperature approaches to the critical point, which represents a smaller temperature variation with the same heat transfer capacity. But the specific heat capacity of water changes in a linear manner. Therefore, the heat transfer process is more difficult because of the possible pinch point. It should be noticed that the heat transfer temperature differences in charge and discharge stages exhibit an opposing trend. Additionally, the heat transfer pinch points typically occur in the middle of WC and at both ends of WH, which should be considered individually in the heat exchanger design process. By and large, it is most advantageous when the specific heat capacities of the two fluids change similarly. Additionally, detailed discrete design of heat exchangers is crucial, particularly when the specific heat capacity of CO₂ varies significantly.

What happens in MHE is a heat exchange process between three fluid streams. The cold fluids include low-temperature methanol and returned gaseous CO₂ and the hot fluid is the CO₂ mainstream. Under the design conditions, the heat transfer temperature difference is maintained at approximately 5 °C and the temperature pinch point locates close to the hot end of MHE. Finally, a straightforward flow and heat transfer process occurs in the MH3, primarily used for cooling methanol.

4.3 Exergy analysis

The exergy efficiency and exergy destruction of each heat exchanger are illustrated in Fig.5, while the relative exergy destruction of each heat exchanger is shown in Fig.6.

The top five heat exchangers with the highest exergy destruction are WH, MH1, MH2, WC, and MHE, accounting for 28.64%, 18.91%, 12.35%, 11.81%, and 7.51% of the total exergy destruction, respectively. This result is consistent with the analysis of heat transfer characteristics above. The available energy loss in WH and WC is primarily attributed to the mismatch of heat transfer curves caused by the significant variation in the specific heat capacity of CO₂. In addition, for MH1 and MC1, the primary cause of energy loss is the gas-liquid phase change of CO₂. Therefore, these heat exchangers require careful design and optimization.

4.4 Sensitivity analysis

The cold storage and liquefaction process based on the L-H cycle is the most complex unit, with many influencing factors to be considered. Therefore, a sensitivity analysis is conducted to examine the impact of key parameters, including liquefaction pressure, liquefaction temperature, and storage pressure on system performance.

In this paper, a typical working condition is established as the analysis basis for analysis, with a liquefaction pressure of 15 MPa, a liquefaction temperature of −25 °C, and a storage pressure of 1 MPa. When one parameter is analyzed, the others are held constant.

First, liquefaction ratio Z is defined in Eq. (27) which represents the proportion of liquid produced after the depressurization process through the TV2.

(27)Z=\frac{m_{\mathrm{l}\mathrm{c}{\mathrm{o}}_2}}{m_{\mathrm{c}{\mathrm{o}}_2}},

where m_{\mathrm{l}\mathrm{c}{\mathrm{o}}_2} represents the mass flow rate of liquid carbon dioxide.

Furthermore, the relationship between Z and RTE are expressed in Eq. (28).

(28)\mathrm{R}\mathrm{T}\mathrm{E}=\frac{W_{\mathrm{T}}}{W_{{\mathrm{C}}_0}(2-Z)+W_{\mathrm{P}}},

in which W_{{\mathrm{C}}_0} represents the compressor power consumption when Z is equal to 1 in the ideal condition. It could be seen that the higher liquefaction ratio is conducive to improving the system RTE.

4.4.1 Liquefaction temperature

The liquefaction temperature refers to the CO₂ temperature before it passes through the TV2. As shown in Fig.7, the proportion of liquid phase decreases with the increase of the liquefaction temperature. The impact of liquefaction temperature on Z and RTE is illustrated in Fig.8.

The liquefaction temperature has a distinct negative impact on the liquefaction ratio and RTE. As the liquefaction temperature rises from −25 to 0 °C, the liquefaction ratio decreases from 91.41% to 76.33% while the RTE decreases from 56.12% to 46.77%.

However, this does not imply that the liquefaction temperature should be set as low as possible, as this would complicate matters and increase costs. Therefore, liquefaction temperature is usually around −25 °C in practical applications.

4.4.2 Liquefaction pressure

Liquefaction pressure is the pressure of CO₂ before the TV2. The throttling process at different liquefaction pressures is shown in Fig.9.

The impact of pressure on liquefaction is not monotonous, which is primarily related to the Joule-Thomson coefficient ({\mu }_{\mathrm{J}\mathrm{T}}) of CO₂. When \mu {}_{\mathrm{J}\mathrm{T}} is greater than 0, the temperature at which the isenthalpy line intersects the saturation line during the throttling process decreases with the increase of liquefaction pressure, resulting in an increase of liquefaction ratio. However, when {\mu }_{\mathrm{J}\mathrm{T}} is less than 0, the temperature mentioned above increases with the increase of liquefaction pressure, resulting in a decrease of liquefaction ratio. In the following analysis, the temperature before throttling is set to −25 °C and the {\mu }_{\mathrm{J}\mathrm{T}} is negative. The impact of liquefaction temperature on liquefaction ratio and RTE is illustrated in Fig.10.

The liquefaction pressure has a limited weak effect on the liquefaction ratio. As the pressure rises from 13 to 18 MPa, the liquefaction ratio increases gradually, remaining consistently close to 91%. However, the RTE exhibits a clear trend of initially rising and then subsequently decreasing. A liquefaction pressure of about 15 MPa maximizes the RTE to 56.12%. This is because both the electricity consumed by the compressor and the amount of compression heat generated during compression increase with the increase of the pressure. The former will increase the CO₂ temperature at the expander inlet, leading to an increase in output work. The increase in expansion work is insufficient to offset the higher compression cost once the pressure exceeds 15 MPa. This combined effect results in an optimal optimal point.

4.4.3 Storage pressure

Storage pressure is equal to the CO₂ pressure after throttling. The throttling process at different storage pressures and the concrete effect are demonstrated in Fig.11 and Fig.12, respectively.

The liquefaction ratio increases with the increase of storage pressure, then the amount of CO₂ that needs to be recompressed decreases. Consequently, input work decreases and RTE improves. However, it is important to note that a high storage pressure can compromise system safety and increase costs.

5 Conclusions

In this paper, a novel LCES system with the L-H liquefaction cycle and two-stage cold/heat storage processes is proposed. Cold/heat storage methods and possible materials are introduced. The heat transfer characteristics of the heat exchangers are analyzed in detail. Additionally, the influence of key parameters on the L-H cycle liquefaction ratio and the RTE is discussed. The main conclusions are summarized as follows:

1) Taking all factors into account, a liquid phase cold storage process based on sensible heat and latent heat separation and a two-stage heat storage scheme which utilizes both thermal oil and pressurized water are designed. Under the the specified design condition, the system achieves an RTE of 56.12%, an ESD of 29.46 kWh/m³, and an exergy efficiency of 93.73%.

2) The top five heat exchangers with the highest exergy destruction are WH, MH1, MH2, WC, and MHE, accounting for 28.64%, 18.91%, 12.35%, 11.81%, and 7.51% of the total exergy destruction, respectively. These heat exchangers require more elaborate structural design and should be prioritized for optimization.

3) There is a positive correlation between the liquefaction ratio and RTE. Therefore, it is necessary to determine the liquefaction pressure, liquefaction temperature, and storage pressure comprehensively based on a comprehensive evaluation of the thermodynamic performance and the practical feasibility of the system.

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