This paper focuses on the coefficient of performance (COP) at maximum figure of merit $χ$ for a Brownian Carnot-like refrigerator, within the context of symmetric Low-Dissipation approach. Our proposal is based on the Langevin equation for a Brownian particle bounded to a harmonic potential trap, which can perform Carnot-like cycles at finite time. We show that under quasistatic conditions the COP has the same expression as the macroscopic Carnot refrigerator. However, for irreversible cycles at finite time and under symmetric dissipation, the optimal COP is the counterpart of Curzon-Ahlborn efficiency for irreversible macroscopic refrigerators.
The present work investigates the generalized extreme bounds of the coefficient of performance (COP) for the power law dissipative Carnot-like refrigerator in the presence of non-adiabatic dissipation under $χ$ and $\dotΩ$ optimization criteria. The lower and upper bounds of the COP for the low dissipation Carnot-like refrigerator under $χ$ and $\dotΩ$ optimization criteria are obtained with power law dissipation level $δ=1$. The comparative analysis of the extreme bounds of COP at optimized $\dot{Ω}$ and $χ$ figure of merit shows the lower bound of the $\dotΩ$ optimized COP is always higher t…
The figure of merit for refrigerators performing finite-time Carnot-like cycles between two reservoirs at temperature $T_h$ and $T_c$ ($<T_h$) is optimized. It is found that the coefficient of performance at maximum figure of merit is bounded between 0 and $(\sqrt{9+8\varepsilon_c}-3)/2$ for the low-dissipation refrigerators, where $\varepsilon_c =T_c/(T_h-T_c)$ is the Carnot coefficient of performance for reversible refrigerators. These bounds can be reached for extremely asymmetric low-dissipation cases when the ratio between the dissipation constants of the processes in contact with the col…
We study the coefficient of performance (COP) and its bounds of the Canot-like refrigerator working between two heat reservoirs at constant temperatures $T_h$ and $T_c$, under two optimization criteria $χ$ and $Ω$. In view of the fact that an "adiabatic" process takes finite time and is nonisentropic, the nonadiabatic dissipation and the finite time required for the "adiabatic" processes are taken into account. For given optimization criteria, we find that the lower and upper bounds of the COP are the same as the corresponding ones obtained from the previous idealized models where any adiabati…
We apply the model of minimally nonlinear irreversible heat engines developed by Izumida and Okuda [EPL {\bf 97}, 10004 (2012)] to refrigerators. The model assumes extended Onsager relations including a new nonlinear term accounting for dissipation effects. The bounds for the optimized regime under an appropriate figure of merit and the tight-coupling condition are analyzed and successfully compared with those obtained previously for low-dissipation Carnot refrigerators in the finite-time thermodynamics framework. Besides, we study the bounds for the nontight-coupling case numerically. We also…
Based on a quantum thermodynamic protocol for shortcut to isothermality that smoothly modify the system-reservoir interaction can significantly speed up an isothermal process while keeping the overall dissipation constant [Phys. Rev. X. 10, 031015 (2020)], we extend the study of optimization performance of Carnot-like heat engines and refrigerators in a straightforward and unified way. We derive the universal optimization efficiency of heat engines and the optimization coefficient of performance of refrigerators under two unified optimization criterions, i.e., chi criterion and omega criterion…
We model a microscopic heat engine as a particle hopping on a one-dimensional lattice in a periodic sawtooth potential, with or without load, assisted by the thermal kicks it gets from alternately placed hot and cold thermal baths. We find analytic expressions for current and rate of heat flow when the engine operates at steady state. Three regions are identified where the model acts either as a heat engine or as a refrigerator or as neither of the two. At quasistatic limit both efficiency of the engine and coefficient of performance of the refrigerator go to that for Carnot engine and Carnot …
At the very foundation of the second law of thermodynamics lies the fact that no heat engine operating between two reservoires of temperatures $T_C\leq T_H$ can overperform the ideal Carnot engine: $\langle W \rangle / \langle Q_H \rangle \leq 1-T_C/T_H$. This inequality follows from an exact fluctuation relation involving the nonequilibrium work $W$ and heat exchanged with the hot bath $Q_H$. In a previous work [Sinitsyn N A, J. Phys. A: Math. Theor. {\bf 44} (2011) 405001] this fluctuation relation was obtained under the assumption that the heat engine undergoes a stochastic jump process. He…
In the present work, a power law dissipative Carnot like heat engine cycle of two irreversible isothermal and two irreversible adiabatic processes with finite time non-adiabatic dissipation is considered and the efficiency under two optimization criteria $\dotΩ$ figure of merit and efficient power, $χ_{ep}$ is studied. The generalized extreme bounds of the optimized efficiency under the above said optimization criteria are obtained. The lower and upper bounds of the efficiency for the low dissipation Carnot-like heat engine under these optimization criteria are obtained with dissipation level …
We study the performance of a three-terminal thermoelectric device such as heat engine and refrigerator with broken time-reversal symmetry by applying the unified trade-off figure of merit ($\dotΩ$ criterion) which accounts for both useful energy and losses. For heat engine, we find that a thermoelectric device working under the maximum $\dotΩ$ criterion gives a significantly better performance than a device working at maximum power output. Within the framework of linear irreversible thermodynamics such a direct comparison is not possible for refrigerators, however, our study indicates that, f…
An implementation of quantum absorption chillers with three qubits has been recently proposed, that is ideally able to reach the Carnot performance regime. Here we study the working efficiency of such self-contained refrigerators, adopting a consistent treatment of dissipation effects. We demonstrate that the coefficient of performance at maximum cooling power is upper bounded by 3/4 of the Carnot performance. The result is independent of the details of the system and the equilibrium temperatures of the external baths. We provide design prescriptions that saturate the bound in the limit of a l…
We consider absorption refrigerators consisting of simultaneously operating Carnot-type heat engine and refrigerator. Their maximum efficiency at given power (MEGP) is given by the product of MEGPs for the internal engine and refrigerator. The only subtlety of the derivation lies in the fact that the maximum cooling power of the absorption refrigerator is not limited just by the maximum power of the internal refrigerator but, due to the first law, also by that of the internal engine. As a specific example, we consider the simultaneous absorption refrigerators composed of low-dissipation (LD) h…
A two-level atomic system as a working substance is used to set up a refrigerator consisting of two quantum adiabatic and two isochoric processes (two constant-frequency processes $ω_a$ and $ω_b$ with $ω_a<ω_b$), during which the two-level system is in contact with two heat reservoirs at temperatures $T_h$ and $T_c (<T_h)$. Considering finite-time operation of two isochoric processes, we derive analytical expressions for cooling rate $R$ and coefficient of performance (COP) $\varepsilon$. The COP at maximum $χ(= \varepsilon R)$ figure of merit is numerically determined, and it is proved to be …
We analytically derive maximum efficiency at given cooling power for Carnot-type low-dissipation refrigerators. The corresponding optimal cycle duration depends on a single parameter, which is a specific combination of irreversibility parameters and bath temperatures. For a slight decrease in power with respect to its maximum value, the maximum efficiency exhibits an infinitely-fast nonlinear increase, which is standard in heat engines, only for a limited range of parameters. Otherwise, it increases only linearly with the slope given by ratio of irreversibility parameters. This behavior can be…
We discuss the effect of subdividing the ratchet potential on the performance of a tiny Brownian heat engine that is modeled as a Brownian particle hopping in a viscous medium in a sawtooth potential (with or without load) assisted by alternately placed hot and cold heat baths along its path. We show that the velocity, the efficiency and the coefficient of performance of the refrigerator maximize when the sawtooth potential is subdivided into series of smaller connected barrier series. When the engine operates quasistatically, we analytically show that the efficiency of the engine can not appr…
While Carnot's model engines demonstrate ideal performances regarding conversion efficiency, they cannot be actually used as energy converters since they are non causal systems. Such an unphysical behavior indeed restrains the working conditions to a single point where, in the case of a refrigerator (generator), the cooling power (output power) vanishes. Focusing on the example of a thermoelectric refrigerator, we study the impact of different dissipation sources on the causality of such systems. Basing our analysis on the block diagram description of this system, we discuss particularly the f…
We show that for systems with broken time-reversal symmetry the maximum efficiency and the efficiency at maximum power are both determined by two parameters: a "figure of merit" and an asymmetry parameter. In contrast to the time-symmetric case, the figure of merit is bounded from above; nevertheless the Carnot efficiency can be reached at lower and lower values of the figure of merit and far from the so-called strong coupling condition as the asymmetry parameter increases. Moreover, the Curzon-Ahlborn limit for efficiency at maximum power can be overcome within linear response. Finally, alway…
We present a detailed study of a Brownian particle driven by Carnot-type refrigerating protocol operating between two thermal baths. Both the underdamped as well as the overdamped limits are investigated. The particle is in a harmonic potential with time-periodic strength that drives the particle cyclically between the baths. Each cycle consists of two isothermal steps at different temperatures and two adiabatic steps connecting them. Besides working as a stochastic refrigerator, it is shown analytically that in the quasistatic regime the system can also act as stochastic heater, depending on …
A microscopic heat engine is modeled as a Brownian particle in a sawtooh potential (with load) moving through a highly viscous medium driven by the thermal kick it gets from alternately placed hot and cold heat reservoirs. We found closed form expression for the current as a function of the parameters characterizing the model. Depending on the values these model parameters take, the engine is also found to function as a refrigerator. Expressions for the efficiency as well as for the refrigerator performance are also reported. Study of how these quantities depend on the model parameters enabled…
We consider a quantum Otto refrigerator cycle of a time-dependent harmonic oscillator. We investigate the coefficient of performance at maximum figure of merit for adiabatic and nonadiabatic frequency modulations. We obtain analytical expressions for the optimal performance both in the high-temperature (classical) regime and in the low-temperature (quantum) limit. We moreover analyze the breakdown of the cooling cycle for strongly nonadiabatic driving protocols and derive analytical estimates for the minimal driving time allowed for cooling.
We study analytically a thermal Brownian motor model and calculate exactly the Onsager coefficients. We show how the reciprocity relation holds and that the determinant of the Onsager matrix vanishes. Such condition implies that the device is built with tight coupling. This explains why Carnot's efficiency can be achieved in the limit of infinitely slow velocities. We also prove that the efficiency at maximum power has the maximum possible value, which corresponds to the Curzon-Alhborn bound. Finally, we discuss the model acting as a Brownian refrigerator.
We study coupled quantum systems as the working media of thermodynamic machines. Under a suitable phase-space transformation, the coupled systems can be expressed as a composition of independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. We also show that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncouple…
We consider the optimization of a finite-time Carnot engine characterized by small dissipations. We bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point, which can be thus chosen optimally. Remarkably, this optimal point is independent of the figure of merit combining power and efficiency that is being maximized. Furthermore, for a general class of dynamics the power output becomes proportional to the heat capacity of the working substance. Since the heat capacity can scale supra-extensively with the number of const…
We model a Brownian heat engine as a Brownian particle that hops in a periodic ratchet potential where the ratchet potential is coupled with a linearly decreasing background temperature. It is shown that the efficiency of such Brownian heat engine is far from Carnot efficiency even at quaistatic limit. At quasistatic limit, the efficiency of the heat engine approaches the efficiency of endoreversible engine $η=1-\sqrt{{T_{c}/T_{h}}}$ \cite{c18}. On the other hand, the maximum power efficiency of the engine approaches $η^{MAX}=1-({T_{c}/T_{h}})^{1\over 4}$. Moreover, the dependence of the curre…
