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Chinese Physics Letters

Quantum Brayton Refrigeration Cycle with Finite-Size Bose–Einstein Condensates

Jiehong 结红 Yuan 袁, Huilin Ruan, Dehua 德华 Liu 刘, Jizhou He, Jianhui 建辉 Wang 王

Published Sep 5, 2023• 8 citations
Publisher

Abstract

We consider a quantum Brayton refrigeration cycle consisting of two isobaric and two adiabatic processes, using an ideal Bose gas of finite particles confined in a harmonic trap as its working substance. Quite generally, such a machine falls into three different cases, classified as the condensed region, non-condensed phase, and regime across the critical point. When the refrigerator works near the critical region, both figure of merit and cooling load are significantly improved due to the singular behavior of the specific heat, and the coefficient of performance at maximum figure of merit is much larger than the Curzon–Ahlborn value. With the machine in the non-condensed regime, the coefficient of performance for maximum figure of merit agrees well with the Curzon–Ahlborn value.

DOI
10.1088/0256-307x/40/10/100502

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We consider a quantum Brayton refrigeration cycle consisting of two isobaric and two adiabatic processes, using an ideal Bose gas of finite particles confined in a harmonic trap as its working substance. Quite generally, such a machine falls into three different cases, classified as the condensed region, non-condensed phase, and regime across the critical point. When the refrigerator works near the critical region, both figure of merit and cooling load are significantly improved due to the singular behavior of the specific heat, and the coefficient of performance at maximum figure of merit is much larger than the Curzon–Ahlborn value. With the machine in the non-condensed regime, the coefficient of performance for maximum figure of merit agrees well with the Curzon–Ahlborn value.
Bose-Einstein condensation is a quintessential characteristic of Bose systems. We investigate the finite-time performance of an endoreversible quantum Brayton heat engine operating with an ideal Bose gas with a finite number of particles confined in a d-dimensional harmonic trap. The working medium of these engines may work in the condensation, noncondensation, and near-critical point regimes, respectively. We demonstrate that the existence of the phase transition during the cycle leads to enhanced engine performance by increasing power output and efficiencies corresponding to maximum power and maximum efficient power. We also show that the quantum engine working across the Bose-Einstein condensation in N-particle Bose gas outperforms an ensemble of independent single-particle heat engines. The difference in the machine performance can be explained in terms of the behavior of specific heat at constant pressure near the critical point regime.
An irreversible model of a quantum refrigeration cycle working with many noninteracting harmonic oscillators is established. The refrigeration cycle consists of two adiabatic and two constant-frequency processes. The general performance characteristics of the cycle are investigated, based on the quantum master equation and the semigroup approach. The expressions for several important performance parameters such as the coefficient of performance, cooling rate, power input, and rate of entropy production are derived. By using numerical solutions, the cooling rate of the refrigeration cycle subject to finite cycle duration is optimized. The maximum cooling rate and the corresponding parameters are calculated numerically. The optimal region of the coefficient of performance and the optimal ranges of temperatures of the working substance and times spent on the two constant-frequency processes are determined. Moreover, the optimal performance of the cycle in the high-temperature limit is compared with that of a classical Brayton refrigerator working with an ideal gas. The results obtained here show that in the high-temperature limit a harmonic quantum Brayton cycle may be equivalent to a classical Brayton cycle.
We explore the quantum version of Brayton cycle with a composite system as the working substance. The actual Brayton cycle consists of two adiabatic and two isobaric processes. Two pressures can be defined in our isobaric process, one corresponds to the external magnetic field (characterized by FxF_xFx​) exerted on the system, while the other corresponds to the coupling constant between the subsystems (characterized by FyF_yFy​). As a consequence, we can define two types of quantum Brayton cycle for the composite system. We find that the subsystem experiences a quantum Brayton cycle in one quantum Brayton cycle (characterized by FxF_xFx​), whereas the subsystem's cycle is of quantum Otto in another Brayton cycle (characterized by FyF_yFy​). The efficiency for the composite system equals to that for the subsystem in both cases, but the work done by the total system are usually larger than the sum of work done by the two subsystems. The other interesting finding is that for the cycle characterized by FyF_yFy​, the subsystem can be a refrigerator while the total system is a heat engine. The result in the paper can be generalized to a quantum Brayton cycle with a general coupled system as the working substance.
A two-level atomic system as a working substance is used to set up a refrigerator consisting of two quantum adiabatic and two isochoric processes (two constant-frequency processes ${\ensuremath{\omega}}_{a}$ and ${\ensuremath{\omega}}_{b}$ with ${\ensuremath{\omega}}_{a}<{\ensuremath{\omega}}_{b})$, during which the two-level system is in contact with two heat reservoirs at temperatures Th{T}_{h}Th​ and Tc(<Th){T}_{c}(<{T}_{h})Tc​(<Th​). Considering finite-time operation of two isochoric processes, we derive analytical expressions for cooling rate RRR and coefficient of performance (COP) $\ensuremath{\varepsilon}$. The COP at maximum $\ensuremath{\chi}(=\ensuremath{\varepsilon}R)$ figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP ${\ensuremath{\varepsilon}}_{\text{CA}}=\sqrt{1+{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}1$, where ${\ensuremath{\varepsilon}}_{C}={T}_{c}/({T}_{h}\ensuremath{-}{T}_{c})$ is the Carnot COP. In the high-temperature limit, the COP at maximum $\ensuremath{\chi}$ figure of merit, ${\ensuremath{\varepsilon}}^{*}$, can be expressed analytically by ${\ensuremath{\varepsilon}}^{*}={\ensuremath{\varepsilon}}_{+}\ensuremath{\equiv}(\sqrt{9+8{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}3)/2$, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within the context of irreversible thermodynamics, we prove that the value of ${\ensuremath{\varepsilon}}_{+}$ is also the upper bound of COP at maximum $\ensuremath{\chi}$ figure of merit when we regard our model as a linear irreversible refrigerator.
We explore the quantum version of the Brayton cycle with a composite system as the working substance. The actual Brayton cycle consists of two adiabatic and two isobaric processes. Two pressures can be defined in our isobaric process; one corresponds to the external magnetic field (characterized by F(x)) exerted on the system, while the other corresponds to the coupling constant between the subsystems (characterized by F(y)). As a consequence, we can define two types of quantum Brayton cycle for the composite system. We find that the subsystem experiences a quantum Brayton cycle in one quantum Brayton cycle (characterized by F(x)), whereas the subsystem's cycle is quantum Otto cycle in another Brayton cycle (characterized by F(y)). The efficiency for the composite system equals to that for the subsystem in both cases, but the work done by the total system is usually larger than the sum of the work done by the two subsystems. The other interesting finding is that for the cycle characterized by F(y), the subsystem can be a refrigerator, while the total system is a heat engine. The result in this paper can be generalized to a quantum Brayton cycle with a general coupled system as the working substance.
The cycle model of a quantum refrigeration cycle working with many non-interacting harmonic oscillators and consisting of two isothermal and two constant-frequency processes is established. Based on the quantum master equation and semi-group approach, the general performance of the cycle is investigated. Expressions for some important performance parameters, such as the coefficient of performance, cooling rate, power input, and rate of the entropy production, are derived. Several interesting cases are discussed and, especially, the optimal performance of the cycle at high temperatures is discussed in detail. Some important characteristic curves of the cycle, such as the cooling rate versus coefficient of performance curves, the power input versus coefficient of performance curves, the cooling rate versus power input curves, and so on, are presented. The maximum cooling rate and the corresponding coefficient of performance are calculated. Other optimal performances are also analysed. The results obtained here are compared with those of an Ericsson or Stirling refrigeration cycle using an ideal gas as the working substance. Finally, the optimal performance of a harmonic quantum Carnot refrigeration cycle at high temperatures is derived easily.