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Communications in Theoretical Physics

Universal Expression of Efficiency at Maximum Power: A Quantum-Mechanical Brayton Engine Working with a Single Particle Confined in a Power-Law Trap*

Zhuolin Ye, Weisheng Li, Yiming Lai, Jizhou He, Jian-Hui 建辉 Wang 王

Published Dec 1, 2015• 2 citations
Publisher

Abstract

Abstract We propose a quantum-mechanical Brayton engine model that works between two superposed states, employing a single particle confined in an arbitrary power-law trap as the working substance. Applying the superposition principle, we obtain the explicit expressions of the power and efficiency, and find that the efficiency at maximum power is bounded from above by the function: η + = θ/(θ + 1), with θ being a potential-dependent exponent.

DOI
10.1088/0253-6102/64/6/671

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Abstract We propose a quantum-mechanical Brayton engine model that works between two superposed states, employing a single particle confined in an arbitrary power-law trap as the working substance. Applying the superposition principle, we obtain the explicit expressions of the power and efficiency, and find that the efficiency at maximum power is bounded from above by the function: η + = θ/(θ + 1), with θ being a potential-dependent exponent.
A Brayton engine is one of the heat engines that convert heat energy into mechanical work. This engine is often implemented on turbojet engines of airplanes and gas turbines. However, the efficiency of a classically Brayton engine still needs to be improved according to the second law of thermodynamics. This research presents a Brayton engine on 2D symmetric potential well. The used method is modeling analogy of the first law of thermodynamics and then it is implemented into a quantum system with the single-particle as a working substance. The primary result obtained is the efficiency formulation of a quantum Brayton engine for 2D symmetric potential well which is similar to the 1D quantum Brayton engine and the classical Brayton engine. Quantum Brayton engine in 2D has a higher efficiency compared to the 1D and the classical one.
A quantum Brayton engine using a massless Boson trapped in a one-dimensional potential well as a working substance has been constructed to be studied. We used a modified analogical model by the first law of thermodynamics for a quantum system implementation. The system was built with Klein Gordon equation, relativistic quantum mechanics Hamiltonian, without the mass term of the Hamiltonian and it was trapped in a one-dimensional potential well. Then, it underwent the quantum ideal Brayton cycle process. The quantum Brayton process consisted of adiabatic compression, isobaric expansion, adiabatic expansion, and isobaric compression processes. This relativistic quantum mechanics Brayton engine has found that the efficiency formulation was similar to the classical one. This study obtained the conformity results between the quantum relativistic and classical Brayton engines. By this cycle process, we also invented that the ratio of heat capacity under constant pressure and volume of the system was 2. It could indicate that the efficiency of quantum relativistic Brayton engine is higher than the classical one.
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.
In their work [J. Phys. A 33, 4427 (2000)], Bender, Brody, and Meister have shown by employing a two-state model of a particle confined in the one-dimensional infinite potential well that it is possible to construct a quantum-mechanical analog of the Carnot engine through changes of both the width of the well and the quantum state in a specific manner. Here, a discussion is developed about realizing the maximum power of such an engine, where the width of the well moves at low but finite speed. The efficiency of the engine at the maximum power output is found to be universal independently of any of the parameters contained in the model.
Quantum Brayton engine efficiency with the massless boson particle as a working substance has been investigated. The research method used is analogy modeling of the first law of classical thermodynamics which is implemented into the quantum systems. The quantum system used is the single massless boson particle trapped inside the two-dimensional symmetrical potential well as a replacement for the classic system is an ideal gas in a piston tube. By replacing the working substance from the classical system into a quantum system, the four processes that form the classic Brayton cycle (two isobaric processes and two adiabatic processes) can be constructed on the quantum system. The final result indicates that the quantum Brayton engine with boson particle has a higher efficiency value compared to the classic version, but is lower than that engine when the massless boson is replaced with nonrelativistic particles. This occurs because of differences in Laplace's constant value in the three systems.