Optimal finite-time Brownian Carnot engine

Recent advances in experimental control of colloidal systems have spurred a revolution in the production of mesoscale thermodynamic devices. Functional “textbook” engines, such as the Stirling and Carnot cycles, have been produced in colloidal systems where they operate far from equilibrium. Simultaneously, significant theoretical advances have been made in the design and analysis of such devices. Here, we use methods from thermodynamic geometry to characterize the optimal finite-time nonequilibrium cyclic operation of the parametric harmonic oscillator in contact with a time-varying heat bath with particular focus on the Brownian Carnot cycle. We derive the optimally parametrized Carnot cycle, along with two other new cycles and compare their dissipated energy, efficiency, and steady-state power production against each other and a previously tested experimental protocol for the Carnot cycle. We demonstrate a 20% improvement in dissipated energy over previous experimentally tested protocols and a $\sim 50\%$ improvement under other conditions for one of our engines, whereas our final engine is more efficient and powerful than the others we considered. Our results provide the means for experimentally realizing optimal mesoscale heat engines.

Figure 1

Shape and temporal parametrizations of a previously tested Brownian Carnot cycle, the optimized Brownian Carnot cycle, and an optimal geodesic cycle matched to the Carnot cycle. (a) The stiffness (top) and temperature (bottom) for three heat engines as a function of time: an experimentally tested Brownian Carnot cycle [10] (solid black), the optimally parametrized Brownian Carnot cycle (dashed orange), and the optimal cycle employing geodesics between adjacent pairs of corners of the previous Carnot cycles (dotted blue). Horizontal axes are shared between subplots. (b) The functional form in control parameter space of the Carnot cycle (solid black) and the optimal geodesic cycle (dotted blue). Orange circles (blue triangles) denote points along the optimal Carnot (geodesic) cycle parametrization at 25 equal duration intervals in time. (c) Same as in (b) for the Clausius diagram of stiffness $k$ against its conjugate thermodynamic force in the quasistatic limit $-X_k=x^2/2=k_{B}T/2k$.

Figure 2

Performance of various engines considered in the main text as a function of protocol duration $\tau$. We numerically simulate the system under cyclic operation of the control parameters for protocol duration $\tau$ and plot the performance of each engine after reaching steady state. In all plots, results of simulations of an engine following the experimental protocol used in Ref. [10] are shown with black squares and a straight black linear interpolation, the optimal Carnot cycle with orange circles and dashed orange linear interpolation, the optimal geodesic cycle with blue triangles and blue dotted linear interpolation, and a hybrid cycle consisting of the optimized Carnot cycle with the cold isothermal compression step replaced by the corresponding geodesic path with green diamonds and dashed linear interpolation. (a) Average power. (b) Average efficiency. (c) Average dissipated energy. (d) Dissipated energy scaled relative to that of experimental protocol. Material parameters, i.e., $m,\zeta ,T_c,T_h$, and $k$, are chosen based on those used in the experiment of Ref. [10]. Observe the dissipation is minimized for the geodesic engine, although this engine is less efficient and less powerful than the optimized Carnot and hybrid engines; the hybrid engine is both most efficient and powerful for all protocol durations considered. (e)–(h) Same as (a)–(d) for different material parameters for which the contrasting performance between different cycles is clearer (seethe Supplemental Material in Ref. [35] for details of material parameter values). Horizontal axes of (a), (c), (e), and (g) are the same as each subfigure below them, (b), (d), (f), and (h), respectively.

Figure 3

The thermodynamic cycles and parametrizations we consider. Closed curves represent the bath temperature ($T$) vs trap stiffness ($k$) for each engine. Markers are placed at 20 points along each cycle separated by equal time intervals. Colors and marker shapes correspond to the conventions in previous figures.