Efficiency at Maximum Power of Low-Dissipation Carnot Engines

Esposito, Massimiliano; Kawai, Ryoichi; Lindenberg, Katja; Broeck, C. Van den

doi:10.1103/physrevlett.105.150603

Cited by 527 publications

(839 citation statements)

References 29 publications

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“…For this reason the notion of efficiency at maximum power has been introduced. An upper bound for the efficiency at maximum power has been proposed long ago by several authors [6][7][8][9] and is commonly referred to as Curzon-Ahlborn upper bound:The range of validity of this bound has been widely discussed in several interesting papers [10][11][12][13][14][15]. For the thermoelectric power generation and refrigeration, within linear response and for systems with time-reversal symmetry, both the maximum efficiency and the efficiency at maximum power, are governed by a single parameter, the dimensionless figure of merit…”

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confidence: 99%

Thermodynamic Bounds on Efficiency for Systems with Broken Time-Reversal Symmetry

Benenti¹,

Saito²,

Casati³

2011

Phys. Rev. Lett.

236

358

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We show that for systems with broken time-reversal symmetry the maximum efficiency and the efficiency at maximum power are both determined by two parameters: a "figure of merit" and an asymmetry parameter. In contrast to the time-symmetric case, the figure of merit is bounded from above; nevertheless the Carnot efficiency can be reached at lower and lower values of the figure of merit and far from the so-called strong coupling condition as the asymmetry parameter increases. Moreover, the Curzon-Ahlborn limit for efficiency at maximum power can be overcome within linear response. Finally, always within linear response, it is allowed to have simultaneously Carnot efficiency and non-zero power. The understanding of the fundamental limits that thermodynamics imposes on the efficiency of thermal machines is a central issue in physics and is becoming more and more practically relevant in the future society. In particular due to the need of providing a sustainable supply of energy and to strong concerns about the environmental impact of the combustion of fossil fuels, there is an increasing pressure to find best thermoelectric materials [1][2][3][4].A cornerstone result goes back to Carnot [5]. In a cycle between two reservoirs at temperatures T 1 and T 2 (T 1 > T 2 ), the efficiency η, defined as the ratio of the performed work W over the heat Q 1 extracted from the high temperature reservoir, is bounded by the so-called Carnot efficiency η C :The Carnot efficiency is obtained for a quasi static transformation which requires infinite time and therefore the extracted power, in this limit, reduces to zero. For this reason the notion of efficiency at maximum power has been introduced. An upper bound for the efficiency at maximum power has been proposed long ago by several authors [6][7][8][9] and is commonly referred to as Curzon-Ahlborn upper bound:The range of validity of this bound has been widely discussed in several interesting papers [10][11][12][13][14][15]. For the thermoelectric power generation and refrigeration, within linear response and for systems with time-reversal symmetry, both the maximum efficiency and the efficiency at maximum power, are governed by a single parameter, the dimensionless figure of meritwhere σ is the electric conductivity, S is the thermoelectric power (Seebeck coefficient), κ is the thermal conductivity, and T is the temperature. The maximum efficiency is given bywhere η C is the Carnot efficiency; the efficiency η(ω max ) at maximum power ω max reads [10]The only restriction imposed by thermodynamics is ZT ≥ 0, so that η max ≤ η C and η(ω max ) ≤ η (l) CA , where η (l) CA = η C /2 is the Curzon-Alhborn efficiency in the linear response regime. The upper bounds η C and η (l) CA are reached when the figure of merit ZT → ∞. This limit corresponds to the so-called strong coupling condition, for which the Onsager matrix L becomes singular (that is, det L = 0) and therefore the ratio J q /J ρ , with J q heat currrent and J ρ electric (particle) current, is independent of the applied temp...

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confidence: 99%

Thermodynamic Bounds on Efficiency for Systems with Broken Time-Reversal Symmetry

Benenti¹,

Saito²,

Casati³

2011

Phys. Rev. Lett.

236

358

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“…Heat engines that attain their maximal power and maximal efficiency (which is either the Carnot efficiency or a lower value) at different working conditions are here defined as heat engines with a power-efficiency trade-off. The power-efficiency trade-off is the subject of many recent studies [3,[6][7][8][9][10][11].Less is known about the efficiency and power of cyclic heat engines, but a lot of research effort has been devoted to understanding them in recent years [12][13][14][15][16][17][18][19]. The operation of a cyclic engine is characterized by a protocol that describes the time dependence of key variables along the cycle -e.g.…”

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confidence: 99%

Geometric Heat Engines Featuring Power that Grows with Efficiency

2016

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Thermodynamics places a limit on the efficiency of heat engines, but not on their output power or on how the power and efficiency change with the engine's cycle time. In this letter, we develop a geometrical description of the power and efficiency as a function of the cycle time, applicable to an important class of heat engine models. This geometrical description is used to design engine protocols that attain both the maximal power and maximal efficiency at the fast driving limit. Furthermore, using this method we also prove that no protocol can exactly attain the Carnot efficiency at non-zero power.Introduction Heat engines -machines that exploit temperature differences to extract useful work, are modeled as operating in either a non-equilibrium steadystate, e.g. thermoelectric [1,2] or chemical potential [3] driven engine, or as a cyclic engine, where external parameters and temperature are varied periodically in time, e.g. the Carnot, Otto, Stirling and the Diesel cycles [4]. Both types of engines are characterized by two main figures of merit: efficiency and power.In steady-state heat engines, currents generated by the temperature difference flow against some affinities (thermodynamical forces), e.g. electrical [1,2,5] or chemical potentials [3], generating useful work. From a practical standpoint, the interest in these engines is limited, since the majority of heat engines are better modeled as cyclic engines. One of the main motivations to study steady-state heat engines, however, is the hope that they share universal characteristics with cyclic engines, which are generically more difficult to analyze. An example for such a characteristic behavior is the relationship between power and efficiency: in all steady state heat engines, the affinity at maximal power does not equal to the affinity at maximal efficiency, unless one of the heat baths is at infinite or zero temperature. Heat engines that attain their maximal power and maximal efficiency (which is either the Carnot efficiency or a lower value) at different working conditions are here defined as heat engines with a power-efficiency trade-off. The power-efficiency trade-off is the subject of many recent studies [3,[6][7][8][9][10][11].Less is known about the efficiency and power of cyclic heat engines, but a lot of research effort has been devoted to understanding them in recent years [12][13][14][15][16][17][18][19]. The operation of a cyclic engine is characterized by a protocol that describes the time dependence of key variables along the cycle -e.g. piston position and temperature. The set of feasible protocols however, is strongly bounded by a set of engine specific and hence non-generic constraints. Maximizing power or efficiency is, therefore, a nontrivial constrained optimization problem. Nevertheless, there is a natural optimization problem in these engines which is both simpler and of practical importance: optimization with respect to overall cycle time. In most cyclic engines, the protocol is determined up to a rescaling of the cycle time. In...

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“…Parallel to spectacular developments in bio-and nanotechnology, there has been great theoretical interest in the study of small-scale machines. A well-documented case is the small-scale Carnot engine, in which the operational unit is subject to thermal fluctuations [1][2][3][4]. Of greater biological relevance are machines that convert one form of work to another, and yet these have received far less attention [5].…”

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Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes

et al. 2012

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We describe a single-level quantum dot in contact with two leads as a nanoscale finite-time thermodynamic machine. The dot is driven by an external stochastic force that switches its energy between two values. In the isothermal regime, it can operate as a rechargeable battery by generating an electric current against the applied bias in response to the stochastic driving and then redelivering work in the reverse cycle. This behavior is reminiscent of the Parrondo paradox. If there is a thermal gradient the device can function as a work-generating thermal engine or as a refrigerator that extracts heat from the cold reservoir via the work input of the stochastic driving. The efficiency of the machine at maximum power output is investigated for each mode of operation, and universal features are identified. Parallel to spectacular developments in bio-and nanotechnology, there has been great theoretical interest in the study of small-scale machines. A well-documented case is the small-scale Carnot engine, in which the operational unit is subject to thermal fluctuations [1][2][3][4]. Of greater biological relevance are machines that convert one form of work to another, and yet these have received far less attention [5]. In this article we introduce an electronic nanodevice that allows several modes of operation. The device is a single-level quantum dot subject to stochastic driving while in contact with two reservoirs that may be at different temperatures and chemical potentials. Its properties can be derived from a stochastic thermodynamic description [6]. We investigate in analytic detail various operational regimes. When operating under tight coupling conditions, familiar features are recovered in appropriate limits: Carnot efficiency for reversible operation when the reservoirs are at different temperatures, universal features of efficiency at maximum power [3,7], and efficiency at maximum power close to the Curzon-Ahlborn efficiency [8]. When the reservoirs are at the same temperature, the work done on the dot by the switching can reverse the "normal" direction (from high to low chemical potential) of the current. Thus, the engine can be seen as a technologically relevant implementation of the Parrondo paradox [9] in that the switching can induce an electron flow against the chemical gradient. When operating under tight coupling, the efficiency at maximum power starts from the universal value of 1/2 close to equilibrium and increases monotonically to 1 as one moves further into the nonequilibrium regime and can thus be much higher than in the traditional implementations of the Parrondo paradox [10]. The same efficiency is observed when the engine works in the reverse mode. I. MODEL AND DYNAMICSWe consider a single-level quantum dot whose energy is stochastically switched between an upper and a lower value, ε j , with j = u or d (Fig. 1). The upward and downward rates are k + and k − . The dot is in contact with a left and a right lead, ν = L or R, at chemical potentials μ ν and temperatures T ν . The transition r...

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Efficiency at Maximum Power of Low-Dissipation Carnot Engines

Cited by 527 publications

References 29 publications

Thermodynamic Bounds on Efficiency for Systems with Broken Time-Reversal Symmetry

Thermodynamic Bounds on Efficiency for Systems with Broken Time-Reversal Symmetry

Geometric Heat Engines Featuring Power that Grows with Efficiency

Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes

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