Universality of efficiency at maximum power

Abstract

We investigate the efficiency of power generation by thermochemical engines. For strong coupling between the particle and heat flows and in the presence of a left-right symmetry in the system, we demonstrate that the efficiency at maximum power displays universality up to quadratic order in the deviation from equilibrium. A maser model is presented to illustrate our argument. The concept of Carnot efficiency is a cornerstone of thermodynamics. It states that the efficiency of a cyclic (''Carnot'') thermal engine that transforms an amount Q ðrÞ of energy extracted from a heat reservoir at temperature T r into an amount of work W is at most ¼ W =Q ðrÞ c ¼ 1 À T l =T r , where T l is the temperature of a second, colder reservoir. The theoretical implications of this result are momentous, as they lie at the basis of the introduction by Clausius of the entropy as a state function. The practical implications are more limited, since the upper limit c (''Carnot efficiency'') is only reached for engines that operate reversibly. As a result, when the efficiency is maximal, the output power is zero. By optimizing the Carnot cycle with respect to power rather than efficiency, Curzon and Ahlborn found that the corresponding efficiency is given by. They obtained this result for a specific model, using in addition the socalled endo-reversible approximation (i.e., neglecting the dissipation in the auxiliary, work producing entity). Subsequently, the validity of this result as an upper bound, as well as its universal character, were the subjects of a long-standing debate. In the regime of linear response, more precisely to linear order in c , it was proven that the efficiency at maximum power is indeed limited by the Curzon-Ahlborn efficiency, which in this regime is exactly half of the Carnot efficiency,. The upper limit is reached for a specific class of models, namely, those for which the heat flux is strongly coupled (i.e., directly proportional) to the work-generating flux. Interestingly, such strong coupling is also a prerequisite for open systems to achieve Carnot efficiency [3,4]. In the nonlinear regime, no general result is known. Efficiencies at maximum power, not only below but also above Curzon-Ahlborn efficiency, have been reported [5][6][7][8]. However, it was also found, again in several strong coupling steady-state models [8][9][10] as well as for an overdamped Brownian particle in a time-dependent harmonic potential [7], that the efficiency at maximum power agrees with CA up to quadratic order in c , i.e., ¼c Þ, again raising the question of universality at least to this order. In this Letter we prove that the coefficient 1=8 is indeed universal for strong coupling models that possess a left-right symmetry. Such a universality is remarkable in view of the fact that most explicit macroscopic relationships, for example, the symmetry of Onsager coefficients, are limited to the regime of linear response. The interest in strong coupling is further motivated by the observation that it can naturally be achieved in nan...

Efficiency at and near maximum power of low-dissipation heat engines

“…The efficiency at maximum Λ reads η = (3 − √ 9 − 8η C )/2. Similarly as the EMP (28), the efficiency at maximum Λ is a monotonically decreasing function of the parameter A (numerical result, see also results shown for the diffusion-based heat engine in Fig. 7).…”

“…This result was further precised [27] and recently a similar argumentation was successfully used for general thermodynamic devices [28]. Further universalities were obtained for the class of heat engines working in the regime of "low dissipation" [15,[29][30][31][32][33], where the work dissipated during the * viktor.holubec@mff.cuni.cz isothermal branches of the Carnot cycle grows in inverse proportion to the duration of these branches.…”

“…In order to determine their values from Eqs. (32)- (33) it is necessary to know the maximum power P ⋆ (27) and the corresponding efficiency η ⋆ (28). While η ⋆ depends solely on the parameters A and η C and thus the efficiency η can be readily obtained, P ⋆ is model-dependent and we calculate it again for the case of a particle diffusing in the logharmonic potential (see Tab.…”

Maximum efficiency of steady-state heat engines at arbitrary power

“…(5)) means that the heat flux is proportional to the flux, which generates work on the surrounding [9][10][11].In the present study we stay in the linear response regime (linear in η C ), however, we go beyond the regime of maximum power and study the engine efficiency at an arbitrary power P , 0 ≤ P ≤ P ⋆ (P ⋆ stands for the maximum power). One of the main messages is that the universal bounds on efficiency can be derived for an arbitrary P and not only at the point of maximum power which was considered in several recent studies [8][9][10][11][12][13][14], see however [15][16][17][18][19][20][21][22] for optimal regimes other than that with maximum power. To this end we introduce relative deviations from the regime of maximum power, the relative gain in efficiency δη and power δP :where −1 ≤ δP ≤ 0.…”

General linear thermodynamics for periodically driven systems with multiple reservoirs

“…Finally, we mention some interesting directions for further research. Firstly, it would be interesting to extend our analysis to higher order response coefficients and to study the resulting constraints on heat engines [31,32]. Secondly, it would be very interesting to see if our results can be extended to quantum mechanical systems and systems with strong coupling [33,34].…”