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

Abstract

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, over-damped stochastic HEs and an under-damped 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.

Maximum efficiency of low-dissipation heat engines at arbitrary power

“…where we have applied the LD assumption (2). The COP measures cost of the cooling in units of input work, W = Q h − Q c , used to pump the heat from the cold bath,…”

“…(5) vanishes. And, even though recent theoretical results on thermodynamics of small systems allowing unprecedented control of the intrinsic relaxation times show that power corresponding to ε C can even diverge [43][44][45][46], it is doomed to be negligible compared to its maximal value [2]. In practice, we thus always have to resort to a compromise between power and COP.…”

“…to show under which conditions is this cooling power cheapest. With respect to heat engines, this task already gained considerable attention in the literature [2]. Expressions for maximum efficiency at given power were derived for quantum thermoelectric heat engines [56,57], LD heat engines [26,58], MNI heat engines [59], a stochastic heat engine based on an uderdamped harmonic oscillator [60], and using general linear response theory [40].…”

“…The loss in power vanishes for P = P and assumes its minimum value −1 if the power P is negligible compared to P [2]. The duration τ equals to −1 for t i = 0 and it is negative (positive) for t i < t * i (t i > t * i ).…”

“…The advantage of these results is their generality. The disadvantages are omnipresent dissipation looses in real machines, rendering their reversible operation difficult, and even more importantly, the fact that reversible conditions correspond to practically negligible output power [2].…”