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arXiv:1408.4917 · Physical Review E

Coefficient of performance under maximum χ criterion in a two-level atomic system as a refrigerator

Yuan Yuan, Rui Wang, Jizhou He, Yongli Ma, Jianhui Wang

Published Nov 25, 2014• 17 citations
arXivHTMLPDF

Abstract

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 ${\ensuremath{\omega}}_{a}$ and ${\ensuremath{\omega}}_{b}$ with ${\ensuremath{\omega}}_{a}<{\ensuremath{\omega}}_{b})$, 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) $\ensuremath{\varepsilon}$. The COP at maximum $\ensuremath{\chi}(=\ensuremath{\varepsilon}R)$ figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP ${\ensuremath{\varepsilon}}_{\text{CA}}=\sqrt{1+{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}1$, where ${\ensuremath{\varepsilon}}_{C}={T}_{c}/({T}_{h}\ensuremath{-}{T}_{c})$ is the Carnot COP. In the high-temperature limit, the COP at maximum $\ensuremath{\chi}$ figure of merit, ${\ensuremath{\varepsilon}}^{*}$, can be expressed analytically by ${\ensuremath{\varepsilon}}^{*}={\ensuremath{\varepsilon}}_{+}\ensuremath{\equiv}(\sqrt{9+8{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}3)/2$, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within the context of irreversible thermodynamics, we prove that the value of ${\ensuremath{\varepsilon}}_{+}$ is also the upper bound of COP at maximum $\ensuremath{\chi}$ figure of merit when we regard our model as a linear irreversible refrigerator.

DOI
10.1103/physreve.90.052151

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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 \ensuremathωa{\ensuremath{\omega}}_{a}\ensuremathωa​ and \ensuremathωb{\ensuremath{\omega}}_{b}\ensuremathωb​ with \ensuremathωa<\ensuremathωb){\ensuremath{\omega}}_{a}<{\ensuremath{\omega}}_{b})\ensuremathωa​<\ensuremathωb​), during which the two-level system is in contact with two heat reservoirs at temperatures Th{T}_{h}Th​ and Tc(<Th){T}_{c}(<{T}_{h})Tc​(<Th​). Considering finite-time operation of two isochoric processes, we derive analytical expressions for cooling rate RRR and coefficient of performance (COP) \ensuremathε\ensuremath{\varepsilon}\ensuremathε. The COP at maximum \ensuremathχ(=\ensuremathεR)\ensuremath{\chi}(=\ensuremath{\varepsilon}R)\ensuremathχ(=\ensuremathεR) figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP \ensuremathεCA=1+\ensuremathεC\ensuremath−1{\ensuremath{\varepsilon}}_{\text{CA}}=\sqrt{1+{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}1\ensuremathεCA​=1+\ensuremathεC​​\ensuremath−1, where \ensuremathεC=Tc/(Th\ensuremath−Tc){\ensuremath{\varepsilon}}_{C}={T}_{c}/({T}_{h}\ensuremath{-}{T}_{c})\ensuremathεC​=Tc​/(Th​\ensuremath−Tc​) is the Carnot COP. In the high-temperature limit, the COP at maximum \ensuremathχ\ensuremath{\chi}\ensuremathχ figure of merit, \ensuremathε∗{\ensuremath{\varepsilon}}^{*}\ensuremathε∗, can be expressed analytically by \ensuremathε∗=\ensuremathε+\ensuremath≡(9+8\ensuremathεC\ensuremath−3)/2{\ensuremath{\varepsilon}}^{*}={\ensuremath{\varepsilon}}_{+}\ensuremath{\equiv}(\sqrt{9+8{\ensuremath{\varepsilon}}_{C}}\ensuremath{-}3)/2\ensuremathε∗=\ensuremathε+​\ensuremath≡(9+8\ensuremathεC​​\ensuremath−3)/2, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within the context of irreversible thermodynamics, we prove that the value of \ensuremathε+{\ensuremath{\varepsilon}}_{+}\ensuremathε+​ is also the upper bound of COP at maximum \ensuremathχ\ensuremath{\chi}\ensuremathχ figure of merit when we regard our model as a linear irreversible refrigerator.
We consider the finite-time operation of a quantum heat engine whose working substance is composed of a two-level atomic system. The engine cycle, consisting of two quantum adiabatic and two quantum isochoric (constant-frequency) processes and working between two heat reservoirs at temperatures Th{T}_{h}Th​ and Tc{T}_{c}Tc​(<(<(<Th){T}_{h})Th​), is a quantum version of the classical Otto cycle. By optimizing the power output with respect to two frequencies, we obtain the efficiency at maximum power output (EMP) and analyze numerically the effects of the times taken for two adiabatic and two isochoric processes on the EMP. In the absence of internally dissipative friction, we find that the EMP is bounded from the upper side by a function of the Carnot efficiency ${\ensuremath{\eta}}_{C}$, ${\ensuremath{\eta}}_{+}={\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})\mathrm{ln}(1\ensuremath{-}{\ensuremath{\eta}}_{C})]$, with ${\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}$. This analytic expression is confirmed by our exact numerical result and is identical to the one derived in an engine model based on a mesoscopic or macroscopic system. If the internal friction is included, we find that the EMP decreases as the friction coefficient increases.
The performance of a quantum Otto refrigerator coupled to a squeezed cold reservoir has been evaluated using the χ figure of merit. We have shown that squeezing can enhance the coefficient of performance (COP) dramatically, surpassing the Carnot COP defined by the initial temperatures of the heat baths. Furthermore, when the squeezing parameter approaches its maximum value, the work input vanishes while the cooling rate remains finite, in apparent contravention of the second law of thermodynamics. To explain this phenomenon, we have shown that squeezing renders the thermal bath into a nonequilibrium state and the temperature of the bath becomes frequency dependent. Thereby, a correlation to the Carnot COP has been deduced. The results reveal that the COP under the maximum χ figure of merit is of the Curzon-Ahlborn style that cannot surpass the actual Carnot COP, and is thus consistent with the second law of thermodynamics.
We consider a quantum Brayton refrigeration cycle consisting of two isobaric and two adiabatic processes, using an ideal Bose gas of finite particles confined in a harmonic trap as its working substance. Quite generally, such a machine falls into three different cases, classified as the condensed region, non-condensed phase, and regime across the critical point. When the refrigerator works near the critical region, both figure of merit and cooling load are significantly improved due to the singular behavior of the specific heat, and the coefficient of performance at maximum figure of merit is much larger than the Curzon–Ahlborn value. With the machine in the non-condensed regime, the coefficient of performance for maximum figure of merit agrees well with the Curzon–Ahlborn value.
We study the optimal performance of a three-level quantum refrigerator using two different objective functions: cooling power and χ function. For both cases, we obtain general expressions for the coefficient of performance (COP) and derive its well-known lower and upper bounds for the limiting cases when the ratio of system-bath coupling constants at the hot and cold contacts approaches infinity and zero, respectively. We also show that the cooling power can be maximized with respect to one control frequency, while χ function can be maximized globally with respect to two control frequencies. Additionally, we show that in the low-temperature regime, our model of refrigerator can be mapped to Feynman's ratchet and pawl model, a classical mesoscopic heat engine. In the parameter regime where both cooling power and χ function can be maximized, we compare the cooling power of the quantum refrigerator at maximum χ function with the maximum cooling power.
We consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures Th{T}_{h}Th​ and Tc{T}_{c}Tc​ (<Th)(<{T}_{h})(<Th​). Although the behavior of spin-1/21/21/2 system differs substantially from that of the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression ${\ensuremath{\eta}}_{\mathrm{mp}}\ensuremath{\le}{\ensuremath{\eta}}_{+}\ensuremath{\equiv}{\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})ln(1\ensuremath{-}{\ensuremath{\eta}}_{C})]$ with ${\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}$ as the Carnot efficiency. This expression ${\ensuremath{\eta}}_{\mathrm{mp}}$ possesses the same universality of the CA efficiency ${\ensuremath{\eta}}_{\mathrm{CA}}=1\ensuremath{-}\sqrt{1\ensuremath{-}{\ensuremath{\eta}}_{C}}$ at small relative temperature difference. Within the context of irreversible thermodynamics, we calculate the Onsager coefficients and show that the value of ${\ensuremath{\eta}}_{\mathrm{CA}}$ is indeed the upper bound of EMP for an Otto engine working in the linear-response regime.
The study of nanoscopic thermal devices attracts an increasing attention, not only because of their key role in understanding the emergence of thermodynamics from quantum theory, but also for the potential use of quantum-mechanical thermal cycles, especially refrigerators, in quantum technologies. This thesis, is devoted to the performance optimization of autonomous (heat-driven) refrigerators. For a quantum refrigerator, one has a full microscopic description of the simple system serving as a working medium, and of its interaction with the external heat baths. This enables a consistent `finite-time? thermodynamic analysis, not relying on any phenomenological assumptions about the leading sources of irreversibility, and allows to establish arguably general performance bounds, essential to benchmark the operation of realistic devices. In particular, tight and model-independent upper bounds for the coefficient of performance at maximum power, are derived here for a class of strongly coupled refrigerators, and seen to hold more generally for other types of heat and power-driven cycles. Remarkably, these ultimate limitations only depend on the particulars of the system-baths interaction at the interface with the object to be cooled. Strategies might be then defined so as to saturate those limits: It is shown how the working medium of the refrigerator may be, for instance, scaled up into a parallel multi-stage configuration, resembling that of macroscopic multi-effect absorption chillers, as a means to increase its cooling power. Similarly, by applying reservoir engineering techniques on selected transitions between the energy states of the working medium, one can mimic exploitable non-equilibrium fluctuations in the baths, capable of boosting the overall power and COP beyond what would be otherwise allowed by the second law of thermodynamics. In this way, heat-driven chillers could be set to compete on a equal footing with the best power-driven refrigerators.