Tight-coupling Condition Research Articles

Nonequilibrium fluctuations of a quantum heat engine

The thermodynamic properties of quantum heat engines are stochastic owing to the presence of thermal and quantum fluctuations. We here experimentally investigate the efficiency and nonequilibrium entropy production statistics of a spin-1/2 quantum Otto cycle in a nuclear magnetic resonance setup. We first study the correlations between work and heat within a cycle by extracting their joint distribution for different driving times. We show that near perfect correlation, corresponding to the tight-coupling condition between work and heat, can be achieved. In this limit, the reconstructed efficiency distribution is peaked at the deterministic thermodynamic efficiency, and fluctuations are strongly suppressed. We further successfully test the second law in the form of a joint fluctuation relation for work and heat in the quantum cycle. Our results characterize the statistical features of a small-scale thermal machine in the quantum domain, and provide means to control them.

Higher-order efficiency bound and its application to nonlinear nanothermoelectrics.

Power and efficiency of heat engines are two conflicting objectives. A tight efficiency bound is expected to give insights on the fundamental properties of such a power-efficiency tradeoff. Here, we derive an upper bound on the efficiency of steady-state heat engines, which incorporates higher-order fluctuations of power. In a prototypical model of nonlinear nanostructured thermoelectrics, we show that the obtained bound is tighter than a well-established efficiency bound derived from the thermodynamic uncertainty relation, demonstrating that the higher-order terms have rich information about the thermodynamic efficiency in the nonlinear regime. In particular, we find that the higher-order bound is exactly achieved if the tight coupling condition is satisfied. The obtained bound gives a consistent prediction with an observation that nonlinearity enhances the power-efficiency tradeoff, and would also be useful in a variety of nanoscale engines.

Dual Resonant Frequency Inductive Power Transfer in an Underwater Tight Coupling System

The application of wireless power transfer technology in the underwater environment proposes both opportunities and challenges to undersea power feeding. Due to the attenuation of electromagnetic waves in seawater, the distance between transmitter and receiver is always maintained at a minimum value, which results in tight coupling between the transmitter and receiver. The tight coupling condition provides a low impedance loop for high-order harmonic, so the component of the harmonic wave is thus significantly increased and cannot be ignored in the power transmission system. In order to fully utilize the harmonic energy, a fundamental-harmonic dual-channel system was proposed and studied in this paper. Compared with single-channel systems transmitting fundamental wave only, the dual-channel system has higher power transmission capability, while the loss in dual channel system can be less than that of a single-channel system after proper optimization. A 3 kW experiment platform is established to verify the effectiveness of theoretical analysis.

Efficiency of a two-stage heat engine at optimal power

We propose a two-stage heat cycle for an optimized linear irreversible heat engine. In the first stage, the heat engine works between the hot reservoir and a finite-sized sink. In the second stage, it works between the finite-sized heat source and cold reservoir. Under the tight-coupling condition, the engine shows the low-dissipation behavior in each stage, i.e., the entropy generated depends inversely on the duration of the process. The phenomenological dissipation constants are determined within the theory itself in terms of the heat transfer coefficients and the heat capacity of the auxiliary system. We study the efficiency at maximum power and highlight a class of efficiencies in the symmetric case that show universality up to second order in Carnot efficiency, while the Curzon-Ahlborn efficiency is obtained as the lower bound for this class.

Attainability of Maximum Work and the Reversible Efficiency of Minimally Nonlinear Irreversible Heat Engines

Abstract We use the general formulation of irreversible thermodynamics and study the minimally nonlinear irreversible model of heat engines operating between a time-varying hot heat source of finite size and a cold heat reservoir of infinite size. We find the criterion under which the optimized efficiency obtained by this minimally nonlinear irreversible heat engine can reach the reversible efficiency under the tight coupling condition: a condition of no heat leakage between the system and the reservoirs. We assume the rate of heat transfer from the hot to the cold heat reservoir obeys Fourier’s law and discuss physical conditions under which one can obtain the reversible efficiency in a finite time with finite power. We also calculate the efficiency at maximum power for the minimally nonlinear irreversible heat engine under the nontight coupling condition.

Nonuniversality of heat-engine efficiency at maximum power

We study the efficiency of a simple quantum dot heat engine at maximum power. In contrast to the quasi-statically operated Carnot engine whose efficiency reaches the theoretical maximum, recent research on more realistic engines operated in a finite time has revealed other classes of efficiencies such as the Curzon-Ahlborn efficiency maximizing the power. Such a power-maximizing efficiency has been argued to be always the half of the maximum efficiency up to the linear order near equilibrium under the tight-coupling condition between thermodynamic fluxes. We show, however, that this universality may break down for the quantum dot heat engine, depending on the constraint imposed on the engine control parameters (local optimization), even though the tight-coupling condition remains satisfied.It is shown that this deviation is critically related to the applicability of the linear irreversible thermodynamics.

Performance analysis of a refrigerator operating between an infinite-sized hot reservoir and a finite-sized cold one within linear irreversible thermodynamics

How to optimize the performances of heat devices operating between finite-sized heat sources and sinks has become a very important issue in the field of finite-time thermodynamics. In this paper, a physical model of the refrigerator operating between an infinite-sized hot reservoir and a finite-sized cold one is proposed, and by using the principles of finite-time thermodynamics and the theory of linear irreversible thermodynamics, we present the analytical expressions of the input power and the coefficient of performance (COP) under the tight-coupling condition, and analyze the performance characteristics of the refrigerator in detail. When the temperature of the cold reservoir is changed with fixing the environment temperature (the temperature of the hot reservoir), it is found that there does not exist a well-defined optimal relation between the input power and a duration time of the refrigerating process, which is a remarkable difference from the working process of a heat engine operating between a finite-sized hot reservoir and an infinite-sized cold one. We further find that the COP exhibits the monotonically decreasing trend with the increase of the input power, but the increase of the exergy leads to the enhancement of the COP. This feature can be understood as follows:when P is small, this means that the duration time is large, thus the refrigerating process approaches to the quasistatic operation, which induces the large COP. In particular, when P0, the COP max. The increase of P implies the reduction of , thus the refrigerating process keeps away from the quasistatic process and approaches to the actual irreversible process, which causes the COP to decrease. On the contrary, shows the increasing trend with the increase of the exergy E. This is because the increase of E means the enhancement of at fixing the input power P, which corresponds to a slow refrigerating process. As a result, E exhibits the increasing behaviors due to the emergence of the quasistatic process. From the above analyses, we can find that an appropriate proposal to optimize the refrigerating performance of heat devices should be based on the actual parameters and the real external environment, thus it is possible to obtain the optimal refrigerating objective at the expense of the suitable input power. These results are not only helpful in the in-depth understanding of the refrigerator operating between an infinite-sized hot reservoir and a finite-sized cold one, but also of great engineering interest in designing realistic heat deices. Our method can also generalize the investigation of heat pumps. In addition, when the tight-coupling condition is false due to the breaking of time-reversal symmetry, there needs to be further consideration about it from the angle of physics.

Endoreversible quantum heat engines in the linear response regime.

We analyze general models of quantum heat engines operating a cycle of two adiabatic and two isothermal processes. We use the quantum master equation for a system to describe heat transfer current during a thermodynamic process in contact with a heat reservoir, with no use of phenomenological thermal conduction. We apply the endoreversibility description to such engine models working in the linear response regime and derive expressions of the efficiency and the power. By analyzing the entropy production rate along a single cycle, we identify the thermodynamic flux and force that a linear relation connects. From maximizing the power output, we find that such heat engines satisfy the tight-coupling condition and the efficiency at maximum power agrees with the Curzon-Ahlborn efficiency known as the upper bound in the linear response regime.

Universality of maximum-work efficiency of a cyclic heat engine based on a finite system of ultracold atoms

We study the performance of a cyclic heat engine which uses a small system with a finite number of ultracold atoms as its working substance and works between two heat reservoirs at constant temperatures Th and Tc(<Th). Starting from the expression of heat capacity which includes finite-size effects, the work output is optimized with respect to the temperature of the working substance at a special instant along the cycle. The maximum-work efficiency ηmw at small relative temperature difference can be expanded in terms of the Carnot value {{boldsymbol{eta }}}_{{boldsymbol{C}}}={bf{1}}-{{boldsymbol{T}}}_{{boldsymbol{c}}}/{{boldsymbol{T}}}_{{boldsymbol{h}}}, {{boldsymbol{eta }}}^{{boldsymbol{m}}{bf{w}}}={{boldsymbol{eta }}}_{{boldsymbol{C}}}/{bf{2}}+{{boldsymbol{eta }}}_{{boldsymbol{C}}}^{{bf{2}}}({bf{1}}/{bf{8}}+{{boldsymbol{a}}}_{{bf{0}}})+{boldsymbol{ldots }}, where a0 is a function depending on the particle number N and becomes vanishing in the symmetric case. Moreover, we prove using the relationship between the temperatures of the working substance and heat reservoirs that the maximum-work efficiency, when accurate to the first order of ηC, reads {{boldsymbol{eta }}}^{{boldsymbol{m}}{bf{w}}}={{boldsymbol{eta }}}_{{boldsymbol{CA}}}+{bf{O}}(ΔT2). Within the framework of linear irreversible thermodynamics, the maximum-power efficiency is obtained as {{boldsymbol{eta }}}^{{boldsymbol{mp}}}={{boldsymbol{eta }}}_{{boldsymbol{CA}}}+{bf{O}}(ΔT2) through appropriate identification of thermodynamic fluxes and forces, thereby showing that this kind of cyclic heat engines satisfy the tight-coupling condition.

Stationary engines in and beyond the linear response regime at the Carnot efficiency.

The condition for stationary engines to attain the Carnot efficiency in and beyond the linear response regime is investigated. We find that this condition for finite-size engines is significantly different from that for macroscopic engines in the thermodynamic limit. For the case of finite-size engines, the tight-coupling condition in the linear response regime directly implies the attainability of the Carnot efficiency beyond the linear response regime. As opposed to this, for the case of macroscopic engines in the thermodynamic limit, there are three types of mechanisms to attain the Carnot efficiency. One mechanism allows engines to attain the Carnot efficiency only in the linear response limit, while the other two mechanisms enable engines to attain the Carnot efficiency beyond the linear response regime. These three mechanisms are classified by introducing a tight-coupling window.

Efficiency and its bounds of minimally nonlinear irreversible heat engines at arbitrary power.

The efficiency for minimally nonlinear irreversible heat engines at any arbitrary power has been systematically evaluated, and general lower and upper efficiency bounds under the tight coupling condition for different operating regions have been proposed, which can be seen as the generalization of the bounds [η_{C}/2<η_{maxP}<η_{C}/(2-η_{C})] on efficiency at maximum power (η_{maxP}), where η_{C} means the Carnot efficiency. We have also calculated the universal bounds of the maximum gain in efficiency in different operating regions to give further insight into the efficiency gain with the power away from the maximum power. In the region of higher loads (higher than the load which corresponds to the maximum power), a small power loss away from the maximum power induces a much larger gain in efficiency. As actual heat engines may not work at the maximum power condition, this paper may contribute to operating actual heat engines more efficiently.

Efficiency at maximum power output for an engine with a passive piston

Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is theoretically studied. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the side wall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call stochastic mean field model.

Optimizing work output for finite-sized heat reservoirs: Beyond linear response.

We uncover an optimization principle for the finite-time heat-work conversion process performed between two finite-sized heat reservoirs in the nonlinear response regime that is characterized by rather generic flux-force relations. We solve the problem of maximizing work output in a given time interval by means of the variational method. Moreover, in the limiting case that the cold reservoir is infinite, we find the corresponding optimized process can be determined by a single quantity, which plays the role similar to that of the Hamiltonian in classical mechanics. Some theoretical implications are discussed consequently, under the generalized tight-coupling condition which applies to both linear and nonlinear response cases. Our results can hopefully help design and control realistic thermodynamical processes.

Linear irreversible heat engines based on local equilibrium assumptions

We formulate an endoreversible finite-time Carnot cycle model based on the assumptions of local equilibrium and constant energy flux, where the efficiency and the power are expressed in terms of the thermodynamic variables of the working substance. By analyzing the entropy production rate caused by the heat transfer in each isothermal process during the cycle, and using the endoreversible condition applied to the linear response regime, we identify the thermodynamic flux and force of the present system and obtain a linear relation that connects them. We calculate the efficiency at maximum power in the linear response regime by using the linear relation, which agrees with the Curzon–Ahlborn (CA) efficiency known as the upper bound in this regime. This reason is also elucidated by rewriting our model into the form of the Onsager relations, where our model turns out to satisfy the tight-coupling condition leading to the CA efficiency.

Four-level refrigerator driven by photons.

We propose a quantum absorption refrigerator driven by photons. The model uses a four-level system as its working substance and couples simultaneously to hot, cold, and solar heat reservoirs. Explicit expressions for the cooling power Q̇(c) and coefficient of performance (COP) η(COP) are derived, with the purpose of revealing and optimizing the performance of the device. Our model runs most efficiently under the tight coupling condition, and it is consistent with the third law of thermodynamics in the limit T→0.

Optimization in finite-reservoir finite-time thermodynamics

A necessary condition to optimize work output is obtained for general heat engines working between two finite-sized heat reservoirs in a given period of time τ, with the amount of heat received from the hot reservoir being fixed for all possible realizations of the process. It states that T(c)σ̇ should be a constant during the optimized process, where T(c) is the temperature of the cold reservoir, which could be time dependent, and σ[over ̇] is the entropy production rate of the two reservoirs. Further optimizing τ gives the maximum time-averaged power output; the corresponding thermodynamical efficiency, however, is found to be generally not equal to half of the maximum efficiency due to the finiteness of the sizes of two reservoirs. Our results are obtained within the framework of Onsager's theory of linear irreversible thermodynamics and under the tight-coupling condition. The findings in this work may potentially be applied in the optimization of realistic thermodynamical processes.

Work output and efficiency at maximum power of linear irreversible heat engines operating with a finite-sized heat source.

We formulate the work output and efficiency for linear irreversible heat engines working between a finite-sized hot heat source and an infinite-sized cold heat reservoir until the total system reaches the final thermal equilibrium state with a uniform temperature. We prove that when the heat engines operate at the maximum power under the tight-coupling condition without heat leakage the work output is just half of the exergy, which is known as the maximum available work extracted from a heat source. As a consequence, the corresponding efficiency is also half of its quasistatic counterpart.

Weighted reciprocal of temperature, weighted thermal flux, and their applications in finite-time thermodynamics

The concepts of weighted reciprocal of temperature and weighted thermal flux are proposed for a heat engine operating between two heat baths and outputting mechanical work. With the aid of these two concepts, the generalized thermodynamic fluxes and forces can be expressed in a consistent way within the framework of irreversible thermodynamics. Then the efficiency at maximum power output for a heat engine, one of key topics in finite-time thermodynamics, is investigated on the basis of a generic model under the tight-coupling condition. The corresponding results have the same forms as those of low-dissipation heat engines [ M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck Phys. Rev. Lett. 105 150603 (2010)]. The mappings from two kinds of typical heat engines, such as the low-dissipation heat engine and the Feynman ratchet, into the present generic model are constructed. The universal efficiency at maximum power output up to the quadratic order is found to be valid for a heat engine coupled symmetrically and tightly with two baths. The concepts of weighted reciprocal of temperature and weighted thermal flux are also transplanted to the optimization of refrigerators.

Coefficient of performance under optimized figure of merit in minimally nonlinear irreversible refrigerator

We apply the model of minimally nonlinear irreversible heat engines developed by Izumida and Okuda (EPL, 97 (2012) 10004) to refrigerators. The model assumes extended Onsager relations including a new nonlinear term accounting for dissipation effects. The bounds for the optimized regime under an appropriate figure of merit and the tight-coupling condition are analyzed and successfully compared with those obtained previously for low-dissipation Carnot refrigerators in the finite-time thermodynamics framework. Besides, we study the bounds for the nontight-coupling case numerically. We also introduce a leaky low-dissipation Carnot refrigerator and show that it serves as an example of the minimally nonlinear irreversible refrigerator, by calculating its Onsager coefficients explicitly.

Onsager coefficients of a Brownian Carnot cycle

We study a Brownian Carnot cycle introduced by Schmiedl and Seifert [Europhys. Lett. 81, 20003 (2008)] from a viewpoint of the linear irreversible thermodynamics. By considering the entropy production rate of this cycle, we can determine thermodynamic forces and fluxes of the cycle and calculate the Onsager coefficients for general protocols, that is, arbitrary schedules to change the potential confining the Brownian particle. We show that these Onsager coefficients contain the information of the protocol shape and they satisfy the tight-coupling condition irrespective of whatever protocol shape we choose. These properties may give an explanation why the Curzon-Ahlborn efficiency often appears in the finite-time heat engines.