Diverging, but negligible power at Carnot efficiency: Theory and experiment

We discuss the possibility of reaching the Carnot efficiency by heat engines (HEs) out of quasistatic conditions at nonzero power output. We focus on several models widely used to describe the performance of actual HEs. These models comprise quantum thermoelectric devices, linear irreversible HEs, minimally nonlinear irreversible HEs, HEs working in the regime of low-dissipation, overdamped stochastic HEs and an underdamped stochastic HE. Although some of these HEs can reach the Carnot efficiency at nonzero and even diverging power, the magnitude of this power is always negligible compared to the maximum power attainable in these systems. We provide conditions for attaining the Carnot efficiency in the individual models and explain practical aspects connected with reaching the Carnot efficiency at large power output. Furthermore, we show how our findings can be tested in practice using a standard Brownian HE realizable with available micromanipulation techniques.

Figure 1

Two major classes of HEs which can achieve the Carnot efficiency. These HEs communicate with reservoirs at temperatures $T_h$ and $T_c$ only. Steady-state HEs [panel (a)] are coupled simultaneously to both reservoirs and operate under time-independent conditions. They transform the difference between the steady heat influx $q_h$ and the steady heat outflow $q_c$ into the output power $P$. Periodically driven HEs coupled always to a single bath at a time [panel (b)] operate in a time-periodic nonequilibrium steady state. These engines accept the total amount of heat $Q_h$ ($-Q_c$) from the hot bath (cold bath) per cycle of the duration $t_p$ and deliver the average output power $P=W/t_p$. Black branches of the cycle are adiabats, and other branches are isotherms.

Figure 2

Upper bound on the efficiency at given power for quantum thermoelectric HEs (dotted black line), for linear irreversible HEs (5) (full blue line), for low-dissipation HEs and minimally nonlinear irreversible HEs (11) (dashed orange line), and for HEs based on an underdamped Brownian particle in a breathing parabola potential (20) (dot-dashed purple line); ${\eta }_C=3/5$.

Figure 3

Sketch of the operational cycle of HEs based on a particle diffusing in a time-dependent potential considered in Secs. 3c, 3d, and 4. The filled blue curve stands for the probability density for the position of the particle. The black parabola represents the potential at the beginning of the individual branches.

Figure 4

Efficiency $\eta$ (upper panel), power $P$ and maximum power $P^{★}$ (middle panel), and two variables (3) measuring the distance from the Carnot efficiency of the engine described in Sec. 4 as functions of the scaling parameter $w_{\infty }$. The increasing efficiency is accompanied with the increasing power in direct contradiction with the quasistatic limit. The detailed description and the parameters used are given in Sec. 4.

Figure 5

Total cycle duration $t_p$ and duration of the cycle at maximum power $t_p^{★}$ (upper panel), minimum and maximum variances of the particle position during the cycle $w_0$ and $w_f$ (middle panel), and the maximum value of the trap stiffness (lower panel) for the engine described in Sec. 4 as functions of the scaling parameter $w_{\infty }$. The region accessible in experiments is roughly $w_{\infty }\in ({10}^3,{10}^4)$, where the cycle time decreases from 1 h to 5 s. A detailed description and the parameters used are given in Sec. 4.

Figure 6

The relative fluctuation of power for the Brownian HE described in Sec. 4 as a function of the scaling parameter $w_{\infty }$. A detailed description and the parameters used are given in Sec. 4.