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Register a new userEfficiency at maximum power: An analytically solvable model for stochastic heat engines
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We study a class of cyclic Brownian heat engines in the framework of finite-time thermodynamics. For infinitely long cycle times, the engine works at the Carnot efficiency limit producing, however, zero power. For the efficiency at maximum power, we find a universal expression, different from the endoreversible Curzon-Ahlborn efficiency. Our results are illustrated with a simple one-dimensional engine working in and with a time-dependent harmonic potential.
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We study the efficiency at maximum power, $eta^*$, of engines performing finite-time Carnot cycles between a hot and a cold reservoir at temperatures $T_h$ and $T_c$, respectively. For engines reaching Carnot efficiency $eta_C=1-T_c/T_h$ in the reversible limit (long cycle time, zero dissipation), we find in the limit of low dissipation that $eta^*$ is bounded from above by $eta_C/(2-eta_C)$ and from below by $eta_C/2$. These bounds are reached when the ratio of the dissipation during the cold and hot isothermal phases tend respectively to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency $eta_{CA}=1-sqrt{T_c/T_h}$ is recovered.
We present a unified perspective on nonequilibrium heat engines by generalizing nonlinear irreversible thermodynamics. For tight-coupling heat engines, a generic constitutive relation of nonlinear response accurate up to the quadratic order is derived from the symmetry argument and the stall condition. By applying this generic nonlinear constitutive relation to finite-time thermodynamics, we obtain the necessary and sufficient condition for the universality of efficiency at maximum power, which states that a tight-coupling heat engine takes the universal efficiency at maximum power up to the quadratic order if and only if either the engine symmetrically interacts with two heat reservoirs or the elementary thermal energy flowing through the engine matches the characteristic energy of the engine. As a result, we solve the following paradox: On the one hand, the universal quadratic term in the efficiency at maximum power for tight-coupling heat engines proved as a consequence of symmetry [M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 102, 130602 (2009); S. Q. Sheng and Z. C. Tu, Phys. Rev. E 89, 012129 (2014)]; On the other hand, two typical heat engines including the Curzon-Ahlborn endoreversible heat engine [F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975)] and the Feynman ratchet [Z. C. Tu, J. Phys. A 41, 312003 (2008)] recover the universal efficiency at maximum power regardless of any symmetry.
We introduce a simple two-level heat engine to study the efficiency in the condition of the maximum power output, depending on the energy levels from which the net work is extracted. In contrast to the quasi-statically operated Carnot engine whose efficiency reaches the theoretical maximum, recent research on more realistic engines operated in finite time has revealed other classes of efficiency such as the Curzon-Ahlborn efficiency maximizing the power output. We investigate yet another side with our heat engine model, which consists of pure relaxation and net work extraction processes from the population difference caused by different transition rates. Due to the nature of our model, the time-dependent part is completely decoupled from the other terms in the generated work. We derive analytically the optimal condition for transition rates maximizing the generated power output and discuss its implication on general premise of realistic heat engines. In particular, the optimal engine efficiency of our model is different from the Curzon-Ahlborn efficiency, although they share the universal linear and quadratic coefficients at the near-equilibrium limit. We further confirm our results by taking an alternative approach in terms of the entropy production at hot and cold reservoirs.
We study the maximum efficiency of a Carnot cycle heat engine based on a small system. It is revealed that due to the finiteness of the system, irreversibility may arise when the working substance contacts with a heat bath. As a result, there is a working-substance-dependent correction to the usual Carnot efficiency, which is valid only when the working substance is in the thermodynamic limit. We derives a general and simple expression for the maximum efficiency of a Carnot cycle heat engine in terms of the relative entropy. This maximum efficiency approaches the usual Carnot efficiency asymptotically when the size of the working substance increases to the thermodynamic limit. Our study extends the Carnots result to cases with arbitrary size working substance and demonstrates the subtlety of thermodynamics in small systems.
Efficiency at maximum power (EMP) is a very important specification for a heat engine to evaluate the capacity of outputting adequate power with high efficiency. It has been proved theoretically that the limit EMP of thermoelectric heat engine can be achieved with the hypothetical boxcar-shaped electron transmission, which is realized here by the resonant tunneling in the one-dimensional symmetric InP/InSe superlattice. It is found with the transfer matrix method that a symmetric mode is robust that regardless of the periodicity, and the obtained boxcar-like electron transmission stems from the strong coupling between symmetric mode and Fabry-Perot modes inside the allowed band. High uniformity of the boxcar-like transmission and the sharp drop of the transmission edge are both beneficial to the maximum power and the EMP, which are optimized by the bias voltage and the thicknesses of barrier and well. The maximum power and EMP are extracted with the help of machine learning technique, and more than 95% of their theoretical limits can both be achieved for smaller temperature difference, smaller barrier width and larger well width. We hope the obtain results could provide some basic guidance for the future designs of high EMP thermoelectric heat engines.
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