A Brayton Pumped Thermal Energy Storage System Based on Supercritical Carbon Dioxide Recompression Reheating Discharge Cycle

You, Dongchuan; Tatli, Akif Eren; Metghalchi, Hameed

doi:10.1115/1.4067446

Abstract

In recent years, renewable energy such as solar energy and large-scale energy storage, which is a very important technology to compensate for solar energy's fluctuation due to weather issues, have been extensively investigated. In this paper, a Pumped Thermal Energy Storage (PTES) cycle based on a supercritical carbon dioxide (sCO₂) Recompression Reheating cycle and energy pump with a recuperator has been proposed and analyzed. Molten salt with varying temperatures of 565 °C to 730 °C has been used for energy storage. The pressure ratios have been fixed in the discharge cycle as 2.5, and in the charging cycle, it varies in order to find the optimum operation condition. Parametric studies have been made to determine the best performance of the new system. Molten salt temperature, split ratio, pressure ratio, and intermediate pressure have been varied in the calculation. Exergy analysis has been developed in order to determine exergy destruction in all components. Roundtrip efficiencies have been calculated over a wide range of operating conditions. Different working fluids such as argon, carbon dioxide, and nitrogen were used in both cycles. Performance was determined for different combinations of working fluids. It is concluded that for best performance working fluid for energy pump (charging) should be argon and carbon dioxide should be the working fluid for discharging cycle. For this combination operating at optimum molten salt temperature, intermediate pressure and split ratio in the discharging cycle, the roundtrip efficiency is 66%, which is the maximum.

1 Introduction

From 1995 to 2015, the global energy demand has a significant increase, about almost 53% in 20 years duration [1]. As per the British Petroleum statistical review in 2021, fossil fuel accounts for almost 83% of global energy consumption [2], which increases emission of greenhouse gases and intense global warming. Renewable energy, as an important part for environment protecting, has been heated discussed recently.

Wind energy and solar energy are two important parts of renewable energy, contributing about 25% and 18% in the total renewable energy in 2023 [3]. The efficiency of wind energy is about 20%–40% [4]. For solar energy, most solar energy production is from solar photovoltaic (PV). In the United States, there are more than 50 PV plants with a registered capacity of equal to or above 100 MW. Another type of solar energy plant is concentrated solar power (CSP) concentrates solar energy on a working fluid to increase its temperature. Currently, the number of CSP plants is about 10 and rising [5]. The efficiency of PV plants is between 15% and 22%, depending on different designs and materials of solar cells. Concentrated solar power (CSP) achieves about 25–30% efficiency [6].

Although CSP does not have a significant advantage in efficiency, its notably consistent output and controllable flexibility [7], have raised much more attention, especially when it combines with thermal energy storage [8]. Concentrated solar power plants could heat the working fluids between 500 and 1000 °C, which is a good fit for the supercritical carbon dioxide (sCO₂) Brayton cycle [9]. Carbon dioxide has been studied as a working fluid in power plants for years due to the favorable high efficiency of the plant using carbon dioxide [10] and environmental safety [11]. Different types of sCO₂ cycles such as reheating cycle [12], intercooling cycle [13], recompression cycle [14,15], turbine split flow cycle [16], and a few others are described in Refs. [17–20]. Tatli et al. [21] analyzed different designs of the sCO₂ Brayton cycle and concluded that the Recompression Reheating (RRH) Brayton has the best performance.

Most renewable energy sources are variable due to weather conditions [22], and so is the solar energy. Thermal energy storage (TES), a system primarily used to store electric energy between two temperature reservoirs, has been used as storage for electric energy for CSP [23]. Compressed Air Energy Storage (CAES) [24], Pumped Hydro Energy Storage (PHES) [25], Liquified Air Energy Storage (LAES), and finally, Pumped Thermal Electricity Storage (PTES) [26] are all technologies for TES.

PTES is the system based on the use of power cycles and Rankine or Brayton energy engines, which have a good fit with CSP using the sCO₂ Brayton cycle for power generation. Pumped Thermal Electricity Storage has been investigated globally and is being commercially developed. Pumped Thermal Electricity Storage has several advantages compared to other electricity storage devices, including no geographical restrictions, long lifetimes, and the ability to use cheap, abundant, non-toxic materials as the storage media [27]. Rindt et al. [28] proposed a Brayton PTES with the highest temperature of 513 °C that achieved 38.9% roundtrip efficiency. Several researchers [28–31] achieved around 60% roundtrip efficiency when 90% isentropic efficiency of the turbine was considered in their calculation.

In this paper, a Brayton Pumped Thermal Energy Storage with a recuperator has been proposed for energy storage in the charging cycle. During the discharge process, a sCO₂ Recompression Reheating cycle is used to transfer energy from molten salt to produce electricity. Parametric studies have been conducted in this article, analyzing how different parameters, such as working fluids, molten salt temperature, split ratio, pressure ratio, and intermediate pressure, affect the performance of the cycle.

2 Model Description

Figure 1 shows the schematic diagram of the Brayton PTES design and sCO₂ power cycle. In the charging process, the energy pump receives electricity for storing and also the working fluid at state 5c receives energy from the sCO₂ plant leaving the energy exchanger at state 6c. Working fluid, then, enters the recuperator, which increases its temperature and enters the compressor at state 1c. After compression at state 2c, it enters the energy exchanger to transfer energy to a molten salt storage tank. Working fluid leaves the energy exchanger at state 3c and goes back to the recuperator. Then, working fluid enters the turbine (state 4c) and leaves at state 5c.

Fig. 1

Schematic diagram for Brayton PTES and sCO2 power cycle

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Schematic diagram for Brayton PTES and sCO₂ power cycle

In the discharge cycle, CO₂ leaves the high-temperature recuperator (HTR), state 4d, and receives energy from the storage tank, which contains high-temperature molten salt leaving at state 5d. The working fluid, then, enters the turbine and leaves at state13d. It is reheated by the storage tank to state 14d. The reheated fluid enters the second turbine and leaves it at state 6d and goes into the HTR, and then enters the low-temperature recuperator (LTR) at state 7d. Carbon dioxide leaves LTR at state 8d, and then, it splits into two parts, states 9d and 10d. Working fluid at state 9d enters the energy exchanger transferring energy to the energy pump when the energy pump is transferring energy to the environment when the energy pump is not in operation. Then, carbon dioxide enters the compressor at state 12d. After compression, state 1d, CO₂ receives energy in the LTR and leaves at state 2d. The CO₂ at state 10d is compressed in another compressor leaving it at state 11d. Carbon dioxide in states 2d and 11d are combined in the mixer leaving at state 3d. The mixture, then, enters the high-temperature recuperator.

3 Working Fluids Properties and Parameters

In this research, the thermodynamic data of carbon dioxide, air, and nitrogen have been determined using REFPROP [32], developed by the National Institute of Standards and Technology (NIST). Properties of argon have been calculated using constant specific heat.

In both cycles, assumptions are made that there is no pressure drop through energy exchangers, mixers, splitter, and recuperators, and the compressors and turbines are all adiabatic. Tables 1 and 2 show values of parameters in the energy pump system and sCO₂ power cycle.

Table 1

Parameters in energy pump system

Parameter	Value
Compressor isentropic efficiency $(η_{c})$	90%
Turbine isentropic efficiency $(η_{t})$	90%
Effectiveness of recuperator $(ε_{r})$	85%
Preheater exit temperature $(T_{6 c})$	T_9d − 10 °C
Molten salt temperature (T_molten)	565–730 °C
High pressure $(p_{high})$	200 bar
Temperature gap (T_2c − T_molten)	10 °C

Parameter	Value
Compressor isentropic efficiency $(η_{c})$	90%
Turbine isentropic efficiency $(η_{t})$	90%
Effectiveness of recuperator $(ε_{r})$	85%
Preheater exit temperature $(T_{6 c})$	T_9d − 10 °C
Molten salt temperature (T_molten)	565–730 °C
High pressure $(p_{high})$	200 bar
Temperature gap (T_2c − T_molten)	10 °C

Table 2

Parameters in supercritical carbon dioxide (sCO₂) power cycle

Discharge parameter	Value
Compressor isentropic efficiency $(η_{c})$	90%
Turbine isentropic efficiency $(η_{t})$	90%
Effectiveness of low-temperature recuperator $(ε_{LTR})$	85%
Effectiveness of high-temperature recuperator $(ε_{HTR})$	85%
Precooler exit temperature $(T_{12 d})$	32 °C
Molten salt temperature (T_molten)	565–730 °C [33]
Low pressure $(p_{low})$	80 bar
High pressure $(p_{high})$	200 bar
Intermediate pressure $(p_{int})$	81–180 bar
Split ratio $(x)$	0.65–0.8
Temperature gap (T_molten − T_14d)	10 °C
Environment temperature (T₀)	20 °C

Discharge parameter	Value
Compressor isentropic efficiency $(η_{c})$	90%
Turbine isentropic efficiency $(η_{t})$	90%
Effectiveness of low-temperature recuperator $(ε_{LTR})$	85%
Effectiveness of high-temperature recuperator $(ε_{HTR})$	85%
Precooler exit temperature $(T_{12 d})$	32 °C
Molten salt temperature (T_molten)	565–730 °C [33]
Low pressure $(p_{low})$	80 bar
High pressure $(p_{high})$	200 bar
Intermediate pressure $(p_{int})$	81–180 bar
Split ratio $(x)$	0.65–0.8
Temperature gap (T_molten − T_14d)	10 °C
Environment temperature (T₀)	20 °C

4 Thermodynamic Model

General energy and entropy balances have been used to determine the performance of the system:

\frac{d}{d t} E_{c v} = {\dot{Q}}_{c v} - {\dot{W}}_{c v} + \sum_{inlets} {\dot{m}}_{i} h_{i} - \sum_{exits} {\dot{m}}_{e} h_{e}

(1)

\frac{d}{d t} S_{c v} = \int (\frac{δ {\dot{Q}}_{j}}{T_{j}}) + \sum_{inlets} {\dot{m}}_{i} s_{i} - \sum_{exits} {\dot{m}}_{e} s_{e} + {\dot{S}}_{i r r}

(2)

For the turbine in the charging process,

η_{t} = \frac{h_{4 c} - h_{5 c}}{h_{4 c} - h_{5 c s}}

(3)

where:

s_{5 c s} = s_{4 c}

(4)

For the compressor:

η_{c} = \frac{h_{2 c s} - h_{1 c}}{h_{2 c} - h_{1 c}}

(5)

and

s_{2 c s} = s_{1 c}

(6)

Using energy balance in the recuperator and definition of effectiveness:

h_{6 c} + h_{3 c} = h_{4 c} + h_{1 c}

(7)

ε_{r} = \frac{h_{1 c} - h_{6 c}}{q_{m a x}}

(8)

q_{max} = min {\begin{matrix} h (T_{3 c}, p_{6 c}) - h_{6 c} \\ h_{3 c} - h (T_{6 c}, p_{3 c}) \end{matrix}

(9)

The network needed in the charging process is:

w_{net} = (h_{2 c} - h_{1 c}) - (h_{5 c} - h_{4 c})

(10)

The energy stored per unit mass of the working fluid is:

q_{moltensalt} = h_{2 c} - h_{3 c}

(11)

The coefficient of performance of the charging process is:

CO P_{charge} = \frac{q_{moltensalt}}{w_{net}}

(12)

The definition of split ratio:

r_{s} = \frac{m_{9 d}}{m_{8 d}}

(13)

According to the definition of compressor efficiency,

η_{c 1} = \frac{h_{1 d s} - h_{12 d}}{h_{1 d} - h_{12 d}}

(14)

where

h_{1 d s}

means enthalpy at the exit assuming the compression process is isentropic:

s_{12 d} = s_{1 d s}

(15)

as it is the same in compressor 2:

η_{c 2} = \frac{h_{11 d s} - h_{10 d}}{h_{11 d} - h_{10 d}}

(16)

where

h_{11 d s}

means enthalpy at the exit assuming the compression process is isentropic:

s_{10 d} = s_{11 d s}

(17)

Energy balance in LTR is:

r_{s} h_{1 d} + h_{7 d} = h_{8 d} + r_{s} h_{2 d}

(18)

According to the definition of recuperator effectiveness:

ε_{LTR} = \frac{r_{s} (h_{2 d} - h_{1 d})}{q_{max}}

(19)

where

q_{max}

here is defined as:

q_{max} = min {\begin{matrix} h_{7 d} - h (T_{1 d}, p_{7 d}) \\ r_{s} (h (T_{7 d}, p_{1 d}) - h_{1 d}) \end{matrix}

(20)

Energy balance in mixer:

r_{s} h_{2 d} + (1 - r_{s}) h_{11 d} = h_{3 d}

(21)

Energy balance in HTR is:

h_{3 d} + h_{6 d} = h_{4 d} + h_{7 d}

(22)

According to the definition of recuperator effectiveness:

ε_{HTR} = \frac{h_{4 d} - h_{3 d}}{q_{max}}

(23)

where

q_{max}

here is defined as:

q_{max} = min {\begin{matrix} h_{6 d} - h (T_{3 d}, p_{6 d}) \\ h (T_{6 d}, p_{3 d}) - h_{3 d} \end{matrix}

(24)

Using turbine efficiency in both turbines:

η_{t 1} = \frac{h_{5 d} - h_{13 d}}{h_{5 d} - h_{13 d s}}

(25)

η_{t 2} = \frac{h_{14 d} - h_{6 d}}{h_{14 d} - h_{6 d s}}

(26)

where

h_{13 d s}

and

h_{6 d s}

mean isentropic exit states and

s_{5 d} = s_{13 d s}

(27)

s_{14 d} = s_{6 d s}

(28)

The network in the discharge process is:

w_{net} = (h_{5 d} - h_{13 d}) + (h_{14 d} - h_{6 d}) - r_{s} (h_{1 d} - h_{12 d}) - (1 - r_{s}) (h_{11 d} - h_{10 d})

(29)

The energy input in the discharge cycle is:

q_{in} = (h_{5 d} - h_{4 d}) + (h_{14 d} - h_{13 d})

(30)

The efficiency of the discharge cycle is:

η_{discharge} = \frac{w_{net}}{q_{in}}

(31)

Or in another way:

η_{discharge} = 1 - \frac{q_{out}}{q_{in}}

(32)

where

q_{out} = r_{s} (h_{9 d} - h_{12 d})

(33)

and the roundtrip efficiency of the whole system is:

ε = \frac{{\dot{W}}_{discharge}^{net}}{{\dot{W}}_{charge}^{net}} = \frac{{\dot{W}}_{discharge}^{net}}{{\dot{Q}}_{in}} \times \frac{{\dot{Q}}_{in}}{{\dot{W}}_{charge}^{net}} = η_{discharge} \times CO P_{charge}

(34)

Since the system operates under steady-state conditions, entropy generation in each component can be calculated:

{\dot{S}}_{irr} = - \int (\frac{δ {\dot{Q}}_{j}}{T_{j}}) - \sum_{inlets} {\dot{m}}_{i} s_{i} + \sum_{exits} {\dot{m}}_{e} s_{e}

(35)

And exergy destruction is:

{\dot{E}}_{x_{d}} = T_{0} {\dot{S}}_{irr}

(36)

5 Results and Discussion

5.1 Charging Cycle.

Different working fluids such as argon, nitrogen, and Carbon dioxide have been used in the charging cycle. Figure 2 shows coefficient of performance (COP) as a function of pressure ratio for the charging cycle using argon as the working fluid. For different molten salt temperatures, COP results stop at different pressure ratios because of the recuperator temperature check to make sure it follows the second thermodynamic law. It can be seen that the COP increases as the Molten Salt (sink of energy) temperature decreases and COP increases as the pressure ratio across the compressor increases. But, the rate of increase of COP for low-pressure ratio across the turbine and compressor is much higher than in the cases where pressure ratios are large. Also, it shows that when the molten salt temperature increases, the performance of the charging cycle decreases.

Fig. 2

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Coefficient of performance as a function of pressure ratio for different molten salt temperatures using Argon as working fluid

Figure 3 shows the performance of the charging cycle while nitrogen is the working fluid. It can be seen that the trend of performance is the same as the system with argon to be the working fluid, but the COP of the nitrogen cycle is less than that of argon to be the working fluid at the same pressure ratio.

Fig. 3

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Coefficient of performance of the charging cycle as a function of pressure ratio having different molten salt temperatures with nitrogen as the working fluid

Figure 4 shows COP of the charging cycle as a function of pressure ratio across the compressor with carbon dioxide as the working fluid. Unfortunately, for carbon dioxide, when its temperature and pressure are close to the critical state, the values of properties become unstable, so at some temperature ranges, and there are no data reported in the diagram. Therefore, in this paper, carbon dioxide will not be considered as the working fluid in the charging cycle.

Fig. 4

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Coefficient of performance of the charging cycle as a function of pressure ratio having different molten salt temperatures with carbon dioxide as the working fluid

Figure 5 shows the coefficient of performances of the charging cycle for three different working fluids of argon, nitrogen, and carbon dioxide. As can be seen, the COP of the charging cycle with argon is the best followed by nitrogen and carbon dioxide. Based on these results, carbon dioxide will not be used as the working fluid for the charging cycle.

Fig. 5

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Coefficient of performance of the charging cycle as a function of pressure ratio with the same molten salt temperature (565 °C) with different working fluids

Figure 6 shows how the coefficient performance of the charging cycle changes with different molten salt temperatures and different working fluids. It can be seen that argon has better performance than nitrogen for all salt temperatures. Therefore, argon is the best choice for the charging cycle, and it has been used in this research.

Fig. 6

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Charging COP versus molten salt temperature with different working fluids

Figure 7 shows exergy destruction in the charging cycle. Exergy destruction percentage is defined as the ratio of exergy destruction in each device over the total exergy destruction in the whole charging cycle. There is no significant change for each component when the molten salt temperature increases. The main exergy destruction in the charging cycle is in the turbine and compressor, which means the isentropic efficiency of the turbine and compressor could affect the COP of the charging cycle significantly.

Fig. 7

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Exergy destruction for each component in charging cycle with different molten salt temperatures

5.2 Discharge Cycle.

The schematic diagram of the discharge cycle is in Fig. 1. It is desired to optimize the performance of the discharge cycle. The thermal efficiency of the discharge cycle is given by Eqs. (31) and (32). For a given molten salt temperature, thermal efficiency is the function of intermediate pressure and split ratio. In this paper, two methods have been used for the determination of optimum intermediate pressure and split ratio. Method 1 uses gradient descent to find the maximum discharge cycle efficiency. Gradient descent is a relatively efficient optimization method discussed by Kingma and Ba in detail [34]. There are many optimizers used for gradient descent, and the Adam optimizer [34] is a much more common way to increase the calculation speed and accuracy by changing the learning rate dynamically when the calculation continues. Detailed algorithms have been discussed in Appendix A.

Method 2 considers using the partial derivatives of thermal efficiency with respect to both split ratio and intermediate pressure. Analytical derivatives from CoolProp were used for enthalpy, while a numerical method was employed to calculate temperature change with respect to these variables. Brent's method [35] (via SciPy's Brent's function) was used to find the roots of partial derivatives of thermal efficiency. The solution method involves two parallel optimization processes:

Optimization with respect to split ratio: (i) For a given source temperature, and intermediate pressure, the optimum split ratio is found by setting the partial derivative of efficiency with respect to the split ratio to zero. (ii) Intermediate pressure and source temperature are then varied to find the global optimum.
Optimization with respect to intermediate pressure: (i) For a given source temperature and split ratio, optimum intermediate pressure is found by setting the partial derivative of efficiency with respect to intermediate pressure to zero. (ii) Split ratio and source temperature are then varied to find the global optimum.

Detailed mathematical formulation and implementation of method 2 are described in Appendix B.

To compare the results of the two methods, Error Percentage (EP), which is defined below, has been calculated. The definition of EP is:

EP = \frac{1}{N} \sqrt{\sum_{i = 1}^{N} {(\frac{A_{i} - B_{i}}{A_{i}})}^{2}}

(37)

where A group are the result of method 1 solution, and B group are the result of method 2 solution. If Error Percentage approached 0, it means the results of the two solution methods are comparable with each other.

Figure 8 shows the optimum intermediate pressure when split ratio and molten salt temperature are changing. The Error Percentage for the two different split ratio results are 2.62 × 10⁻⁴ and 1.51 × 10⁻⁵, which shows the perfect agreement between two methods.

Fig. 8

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Optimum intermediate pressure with different split ratios and as a function molten salt temperature

Figure 9 shows the optimum split ratio for different intermediate pressures as a function of molten salt temperature. It can be seen in the figure that two results are very close to each other. The Error Percentage for the two different results are 2.89 × 10⁻⁵ and 2.34 × 10⁻⁵, which also proves that the results of the two methods are very close to each other.

Fig. 9

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Optimum split ratio for different pressure ratios as a function of molten salt temperature

Figure 10 shows the optimum discharge cycle thermal efficiency using different working fluids: carbon dioxide, argon, and air. It can be seen that the supercritical CO₂ cycle has a better performance than Argon and air. Therefore, in the discharge cycle, carbon dioxide has been chosen as the working fluid.

Fig. 10

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Optimum discharge cycle thermal efficiency for different working fluids as a function of molten salt temperature

Exergy analysis have been made for the discharge cycle, and Table 3 shows the exergy destruction percentage in each component in the discharge cycle when the molten salt temperature is 565 °C. Exergy destruction percentage is defined as the ratio of exergy destruction in each device over the total exergy destruction in the whole discharge cycle. The largest exergy destruction takes place in the recuperators, which is much higher than turbines and compressors. Figure 11 shows the percentage of exergy destruction of the two recuperators, HTR and LTR. The total exergy destruction in these two recuperators is over 50% of the total exergy destruction. This high percentage of exergy destructions in the two recuperators could be the basis of research for energy exchanger designers. Figure 12 illustrates that the exergy destruction attributed to the two turbines and two compressors accounts for approximately 15% of the total exergy destruction in the system.

Fig. 11

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Exergy destruction percentage for two recuperators in discharge cycle with different molten salt temperatures

Fig. 12

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Exergy destruction percentage for compressors and turbines in discharge cycle with different molten salt temperatures

Table 3

Exergy destruction ratio in different components in discharge cycle when the molten salt temperature is 565 °C

Discharge device	Exergy destruction percentage
HTR	30.45%
LTR	21.02%
Energy exchanger 1	12.66%
Turbine 2	6.44%
Turbine 1	4.14%
Compressor 2	2.31%
Compressor 1	2.27%
Energy exchanger 2	1.53%
Mixer	0.50%

Figure 13 shows the optimum thermal efficiency for the discharge cycle as a function of the molten salt temperature. In this figure, the results of two methods have been shown. The Error Percentage of the two methods is 1.52 × 10⁻⁵, which is small enough to prove that the two results are close. In the figure, the optimum thermal efficiency increases as molten salt temperature increases.

Fig. 13

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Optimum thermal efficiency with different molten salt temperatures

5.3 Roundtrip Efficiency.

Figure 14 shows optimum roundtrip efficiency as a function of molten salt temperature while using argon in the charging cycle and carbon dioxide in the discharge cycle for different temperature differences. The temperature difference $(Δ T)$ is defined as the temperature difference between states 9d and 6c of the charging cycle. It can be seen that the roundtrip efficiency increases as the molten salt temperature increases. But, the increase in roundtrip efficiency is a very weak function of molten salt temperature. The figure shows for a 180 °C increase of molten salt temperature the roundtrip efficiency only increases by 1%. The same conclusion is also true for the temperature differences.

Fig. 14

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Roundtrip efficiency with different molten salt temperatures

6 Conclusion

In this paper, a new combination of charging and discharging cycles has been proposed. The charging cycle uses a Brayton Pumped Thermal Energy Storage cycle using argon as working fluid, and a supercritical carbon dioxide (sCO₂₎ Recompression Reheating cycle has been proposed for the discharge cycle. A thorough parametric study has been done, and the conclusions are as follows:

Coefficient of performance of the charging cycle decreases as the molten salt energy storage temperature increases.
Two different numerical methods have been used in determining the optimum performance of the discharge cycle, and it has been shown that both methods predict the same results.
The analysis shows that the thermal efficiency of the discharge cycle increases as molten salt temperature increases.
Roundtrip efficiency of the whole system is a very weak function of the temperature of the molten salt storage tank. It increases by 1% for the given range of molten salt temperature.
The optimum operation condition shows a roundtrip efficiency of 66%.
Using exergetic analysis, the largest exergy destruction occurs in low-temperature recuperator and high-temperature recupeartor in the discharge cycle, which means increasing recuperator effectiveness could improve the performance of the discharge cycle. For the charging cycle, the most exergy destruction takes place in turbine and compressor.

Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent was obtained for all individuals. Documentation is provided upon request. This article does not include any research in which animal participants were involved.

Data Availability Statement

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

Appendix A

The goal of optimization is to find the maximum discharge thermal efficiency with respect to split ratio (r_s) and intermediate pressure (P_int). Efficiency could be treated as a function of r_s and P_int:

η_{discharge} = \frac{w_{net}}{q_{in}} = η (r_{s}, P_{int})

(A1)

Since gradient descent could only find the minimum, it is required to reverse the function. Also, pint is much larger than r_s, so a normalization is needed, and our target function is:

f (r_{s}, P_{int}^{*}) = - η (r_{s}, P_{int}^{*})

(A2)

where pint* is the normalization of pint.

First order, or gradient, is required in gradient descent. A numerical approach is followed by the equation below:

f^{'} (x) \approx \frac{f (x + d x) - f (x - d x)}{2 d x}

(A3)

The value of dx is 10⁻⁶ to make the derivative accurate enough.

Adam optimizer is fulfilled in the gradient descent, since it is dynamic learning rate (step size) makes the calculation faster and more accurate than other optimizers.

A short outline [34] for the calculation process is:

\begin{aligned} while u = [r_{s}, P_{int}^{*}] not converged \\ t = t + 1; g_{t} = \nabla_{u} f_{t} (u_{t} - 1) \\ m_{t} = β_{1} * m_{t - 1} + (1 - β_{2}) * g_{t} \\ v_{t} = β_{2} * v_{t - 1} + (1 - β_{2}) * g_{t}^{2} \\ {\bar{m}}_{t} = \frac{m_{t}}{1 - β_{1}^{t}} \\ {\bar{v}}_{t} = \frac{v_{t}}{1 - β_{2}^{t}} \\ u_{t} = u_{t - 1} - α * \frac{{\bar{m}}_{t}}{(\sqrt{\bar{v_{t}}} + ϵ)} \\ end while \\ return u_{t} \end{aligned}

In the calculation process, t means timestep, or iteration times. Some parameters' initial values [34] are listed below:

\begin{aligned} β_{1} = 0.9, β_{2} = 0.999, ϵ = 10^{- 8}, α = 0.001 \\ m_{0} = 0, v_{0} = 0, t = 0 \end{aligned}

Appendix B

Using Eq. , the thermal efficiency of the recompression reheating cycle is defined as:

\begin{matrix} η = 1 - \frac{q_{out}}{q_{in}} \end{matrix}

(B1)

where

q_{out}

is the energy rejection by heat interaction from precoolers and

q_{in}

is the energy input by heat interaction to intermediate energy exchangers.

\begin{matrix} q_{out} = f = r_{s} (h_{9 d} (T, P_{l}) - h_{12 d} (T_{12 d}, P_{l})) \end{matrix}

(B2)

\begin{matrix} q_{in} = g = (h_{5 d} (T_{5 d}, P_{h}) - h_{4 d} (T, P_{h})) + (h_{14 d} (T_{14 d}, P_{int}) - h_{13 d} (T, P_{int})) \end{matrix}

(B3)

Here, $r_{s}$ is the split ratio and $h_{i}$ represents the specific enthalpy at state point i in the cycle, $T_{5 d}$ and $T_{14 d}$ are constant turbine inlet temperatures, $T_{12 d}$ is constant main compressor inlet temperature, and T represents the variable temperature of a state depending on split ratio and intermediate pressure. $P_{h}, P_{l}, and P_{int}$ are high pressure, low pressure, and intermediate pressure, respectively.

Enthalpy is a function of temperature and pressure:

\begin{matrix} h = h (T, P) \end{matrix}

(B4)

Derivative of enthalpy is given by:

\begin{matrix} d h = {(\frac{\partial h}{\partial T})}_{P} d T + {(\frac{\partial h}{\partial P})}_{T} d P \end{matrix}

(B5)

Partial derivative of enthalpy with respect to the split ratio for constant pressures is calculated:

\begin{matrix} \frac{\partial h}{\partial r_{s}} = {(\frac{\partial h}{\partial T})}_{P} (\frac{d T}{d r_{s}}) \end{matrix}

(B6)

Partial derivative of enthalpy with respect to intermediate pressure is calculated:

\begin{matrix} \frac{\partial h}{\partial P_{int}} = {(\frac{\partial h}{\partial T})}_{P_{int}} (\frac{d T}{d P_{int}}) + {(\frac{\partial h}{\partial P_{int}})}_{T} \end{matrix}

(B7)

When the pressure of the state is constant, the derivative of enthalpy with respect to intermediate pressure is calculated:

\begin{matrix} \frac{\partial h}{\partial P_{int}} = {(\frac{\partial h}{\partial T})}_{P} (\frac{d T}{d P_{int}}) \end{matrix}

(B8)

Temperature derivatives with respect to

r_{s}

and

P_{int}

are calculated numerically using central difference approximation. The central finite difference formula for the change of temperature with respect to split ratio and intermediate pressure is given by:

\begin{matrix} \frac{d T}{d x} = \frac{T (x + x) - T (x - x)}{2 x} \end{matrix}

(B9)

where x represents $r_{s}$ or $P_{int}$ ⁠, and x is a small step size for the change in these variables.

Partial derivative of efficiency with respect to

r_{s}

is given by:

\begin{matrix} \frac{\partial η}{\partial r_{s}} = - \frac{1}{g^{2}} (\frac{\partial f}{\partial r_{s}} g - f \frac{\partial g}{\partial r_{S}}) \end{matrix}

(B10)

where

\begin{matrix} \frac{\partial f}{\partial r_{s}} = r_{s} \frac{\partial h_{9 d}}{\partial r_{s}} + (h_{9 d} - h_{12 d}) \end{matrix}

(B11)

\begin{matrix} \frac{\partial g}{\partial r_{S}} = - \frac{\partial h_{4 d}}{\partial r_{s}} - \frac{\partial h_{13 d}}{\partial r_{s}} \end{matrix}

(B12)

Similarly, the partial derivative with respect to

P_{int}

is

\begin{matrix} \frac{\partial η}{\partial P_{int}} = - \frac{1}{g^{2}} (\frac{\partial f}{\partial P_{int}} g - f \frac{\partial g}{\partial P_{int}}) \end{matrix}

(B13)

where

\begin{matrix} \frac{\partial f}{\partial P_{int}} = r_{s} \frac{\partial h_{9 d}}{\partial P_{int}} \end{matrix}

(B14)

\begin{matrix} \frac{\partial g}{\partial P_{int}} = - \frac{\partial h_{4 d}}{\partial P_{int}} - \frac{\partial h_{13 d}}{\partial P_{int}} + \frac{\partial h_{14 d}}{\partial P_{int}} \end{matrix}

(B15)

Finally, the equations to be solved are:

\frac{\partial η}{\partial r_{s}} = - \frac{1}{q_{i n}^{2}} (\begin{matrix} ((r_{s}) \begin{matrix} (\frac{\partial h_{9 d}}{\partial T_{9 d}}) |_{P_{l}} \end{matrix} \frac{d T_{9 d}}{d r_{s}} + (h_{9 d} - h_{12 d})) q_{in} - \\ ({- (\frac{\partial h_{4 d}}{\partial T_{4 d}}) |}_{P_{h}} \frac{d T_{4 d}}{d r_{s}} - {(\frac{\partial h_{13 d}}{\partial T_{13 d}}) |}_{P_{int}} \frac{d T_{13 d}}{d r_{s}}) q_{out} \end{matrix})

(B16)

\frac{\partial η}{\partial P_{int}} = - \frac{1}{q_{in}^{2}} (\begin{matrix} ((r_{s}) {(\frac{\partial h_{9 d}}{\partial T_{9 d}}) |}_{P_{l}} \frac{d T_{9 d}}{d P_{int}}) q_{in} - \\ (- {(\frac{\partial h_{4 d}}{\partial T_{4 d}}) |}_{P_{h}} {\frac{d T_{4 d}}{d P_{int}} + \frac{\partial h_{14 d}}{\partial P_{int}} |}_{T_{14 d}} - {{\frac{\partial h_{13 d}}{\partial T_{13 d}} |}_{P_{int}} \frac{d T_{13 d}}{d P_{int}} + \frac{\partial h_{13 d}}{\partial P_{int}} |}_{T_{13 d}}) q_{out} \end{matrix})

(B17)

To find the optimum

r_{s}

value, we solve:

\begin{matrix} \frac{\partial η}{\partial r_{s}} = 0 \end{matrix}

(B18)

This is found numerically using Brent's method which was implemented via the brentq function from SciPy.

Similarly, to find the optimum

P_{int}

value, we solve:

\begin{matrix} \frac{\partial η}{\partial P_{int}} = 0 \end{matrix}

(B19)

This is also solved using Brent's method.

Global optimization process involves iterating over a range of $T_{s}$ ⁠:

For each $T_{s}$ ⁠:

A range of $P_{int}$ values was explored finding the optimum $r_{s}$ for each $P_{int}$ ⁠.
Also, a range of $r_{s}$ values was explored finding the optimum $P_{int}$ for each $r_{s}$ ⁠.

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A Brayton Pumped Thermal Energy Storage System Based on Supercritical Carbon Dioxide Recompression Reheating Discharge Cycle

Abstract

1 Introduction

2 Model Description

3 Working Fluids Properties and Parameters

4 Thermodynamic Model

5 Results and Discussion

5.1 Charging Cycle.

5.2 Discharge Cycle.

5.3 Roundtrip Efficiency.

6 Conclusion

Conflict of Interest

Data Availability Statement

Appendix A

Appendix B

References

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A Brayton Pumped Thermal Energy Storage System Based on Supercritical Carbon Dioxide Recompression Reheating Discharge Cycle

Abstract

1 Introduction

2 Model Description

3 Working Fluids Properties and Parameters

4 Thermodynamic Model

5 Results and Discussion

5.1 Charging Cycle.

5.2 Discharge Cycle.

5.3 Roundtrip Efficiency.

6 Conclusion

Conflict of Interest

Data Availability Statement

Appendix A

Appendix B

References

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