Enhancing Otto-mobile Efficiency via Addition of a Quantum Carnot Cycle

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1 Fortschr. Phys. 50 (00) 5 7, Enhancing Otto-mobile Efficiency via Addition of a Quantum Carnot Cycle omáš Opatrný and Marlan O. Scully Department of Physics, exas A & M University, College Station, X 778, USA Abstract It was shown recently that one can improve the efficiency of the Otto cycle by taking advantage of the internal degrees of freedom of an ideal gas [M. O. Scully, he Quantum Afterburner, Phys. Rev. Lett., to be published]. Here we discuss the limiting improvement of the efficiency by considering reversible cycles with both internal and external degrees of freedom.. Introduction Quantum thermodynamics, a field opened by Planck, lies at the heart of the old quantum theory. he new quantum mechanics, i.e., matrix and wave mechanics as developed by Heisenberg and Schrödinger, is currently being applied to thermodynamics. his new take on quantum thermodynamics is yielding interesting new insights. For example, the quantum heat engine idea by Scovil and Schulz-DuBois [] and various extensions of this work [] have contributed to better understanding of the thermodynamic concepts from the quantum mechanical point of view. A recent example is the demonstration of the Otto cycle efficiency improvement by extracting useful laser work from the internal degrees of freedom [, ]. Here we discuss the limiting case of such an improvement, illustrating it by a simple model as in Fig.. he Otto cycle consists of two adiabatic and two isochoric processes (see Fig. ) and its efficiency h O is determined by the volume span h O ¼ V g g V ðþ (g being the gas constant), or equivalently, by the temperature span of the hotter adiabatic process h O ¼ : ðþ Interestingly, the efficiency does not depend on the temperatures of the colder adiabatic process, in particular, on the coldest temperature of the cycle. It was suggested recently [, ] to take advantage of the internal degrees of freedom of the gas particles (which are considered to be completely decoupled from the center of mass (COM) degrees of freedom) to improve the Otto cycle efficiency. A combined system of masers and lasers working between the hottest and coldest temperatures of the cycle can extract useful laser work, thus increasing the overall efficiency. # WILEY-VCH Verlag Berlin GmbH, 086 Berlin, /0/ $ 7.50þ.50/0

2 658. Opatrný and M. O. Scully, Enhancing Otto-mobile Efficiency W (Carnot) Magnet B W Heat reservoir (Otto) Fig.. Combined Otto and Carnot cycle. he Otto cycle works with the COM degrees of freedom and produces work W ðottoþ, while the Carnot cycle works with the internal magnetic levels of the medium. A permanent magnet producing field B is attracted to the paramagnetic medium. he magnet moves towards the medium when the medium is cold and the attractive force is large, and it moves in the opposite direction when the medium is hot and the attractive force is small. As a result, net work W ðcarnotþ is produced. We will show that a simple Carnot cycle operating on the internal degrees of freedom between the hottest and coldest temperatures can increase the Otto cycle efficiency by the maximum amount. In particular, if each of the gas particles has N internal states decoupled from the COM and capable of a separate thermodynamic cycle, then the combined cycle efficiency can approach the limiting value þ ðg Þ ln N h max ¼ þ : ðþ ðg Þ ln N An example (see Fig. ) of such an improved cycle is discussed.. Basic Considerations of the Otto Cycle Let us consider the cycle as in Fig.. he heat received by the working fluid from the hot reservoir during the isochoric heating with the smaller volume V is ¼ C V ð Þ ; ðþ whereas the heat transmitted to the cold reservoir during the isochoric cooling with the bigger volume V is ¼ C V ð Þ : ð5þ he net work produced by the working fluid is the difference between the heat input and output, W ðottoþ ¼. Since from the adiabatic processes we have ¼ V g g ¼ ; ð6þ V

3 Fortschr. Phys. 50 (00) we can write for the work W ðottoþ ¼ C V ð Þ : ð7þ Using the definition of the heat-engine efficiency h O ¼ W ðottoþ ; ð8þ we obtain the expressions () and (). As can be seen, lowering the coldest temperature does not improve the Otto cycle efficiency, although it helps in increasing the net work W ðottoþ.. Carnot Cycle with an N Level System Let us consider first a two-level system, the generalization to an N level system being straightforward. Let us assume that one can change the energy difference of the two levels by an external parameter (e.g., if the levels refer to magnetic momentum states, their energies are determined by the external magnetic field). he Carnot cycle then looks as follows (see Fig. ): State : the energy difference of the levels is E, the temperature is. he entropy of the state is S ¼ SðE ; Þ, where the entropy of a two-level system is given by SðE; Þ ¼k x E k ; ln ð þ e x Þþ x e x þ where k is the Boltzmann constant. S P Fig.. S and PV diagrams for an ideal Otto cycle V V V ; ð9þ ð0þ ε ε ε ε Adiabat. Isotherm. Adiabat. Isotherm. Fig.. A scheme of a Carnot cycle with a two-level system

4 660. Opatrný and M. O. Scully, Enhancing Otto-mobile Efficiency Isothermal heating: the energy difference is being decreased while heat flows from the hot reservoir to the system. he heat given to the system is ¼ ½SðE ; Þ SðE ; ÞŠ : ðþ State : the energy difference is E, the temperature remains. he entropy is S ¼ SðE ; Þ. Adiabatic decrease of the energy difference: the population of the levels remains constant, and because the energy difference is decreasing, the temperature decreases, too. State : the energy difference is E, the temperature is (taken to be equal to the lowest temperature of the Otto cycle). he entropy is S ¼ SðE ; Þ¼S. Isothermal cooling: the energy difference increases and the temperature is kept low by contact with the reservoir. he heat extracted from the system is ¼ ½SðE ; Þ SðE ; ÞŠ: ðþ State : the energy difference is E and the temperature is. E is chosen such that the entropy is S ¼ SðE ; Þ¼S. Adiabatic increase of the energy difference: the system returns to the initial state. he net work performed by the system is W ðcarnotþ ¼ : ðþ If the temperatures and are fixed, the maximum work is obtained if the system energy differences are chosen such that and One then obtains E ; E k! SðE ; Þ ; SðE ; Þk ; ðþ E ; E k! SðE ; Þ ; SðE ; Þk ln : ð5þ k ln ; ð6þ k ln ; ð7þ from which the maximum extractable work is W ðcarnotþ max ¼ kð Þ ln : ð8þ It is not difficult to generalize the considerations to an N level system, for which W ðcarnotþ max ¼ kð Þ ln N : ð9þ A Carnot engine working with internal degrees of freedom can be realized using, e.g., magnetic levels of a paramagnetic substance in an external magnetic field (see the S and mb

5 Fortschr. Phys. 50 (00) m S Fig.. S and mb diagrams for a Carnot cycle with a two-level magnetic system. he quantities m and B refer to the magnetic moment and the external magnetic field, respectively. Note that during an adiabatic process the mean magnetic moment of a two-level system remains constant. diagrams in Fig. ). Similar cycles are used in cooling devices [5, 6] to reach temperatures about K. Magnetic Carnot cycles with mechanical output work were studied, e.g., in [7]. In our model the Carnot cycle produces mechanical work by moving a permanent magnet in vicinity of the paramagnetic working fluid (see Fig. ). he magnet moves towards the fluid when the fluid is cold and the magnetic moments of its particles are mostly aligned with the magnetic field. hus the magnetization of the fluid is large and useful work is produced by moving the magnet in the direction of the large attractive force. he magnet moves in the opposite direction when the fluid is hot and many magnetic moments point against the external field due to the thermal motion. he magnetization of the fluid is then small so that the waste work necessary for moving the magnet against the force is smaller than the useful work. Alternately, the magnetic Carnot cycle can produce electric work, e.g., if the working fluid is used as a core of a coil in an LC oscillator. Periodic connection of the paramagnetic fluid to the heat and cold reservoirs then induce parametric oscillations in the circuit. B. Combined Cycle Let us consider a combined cycle with the COM degrees of freedom used to produce mechanical work in an Otto cycle, and the internal levels are used separately in a Carnot cycle working between the maximum an minimum temperatures and. he individual steps of the cycle are (cf. []): ð! Þ he hot gas expands adiabatically, doing useful work. COM temperature drops from to, internal temperature remains (same as in []). ð! Þ Heat is extracted from the COM at constant volume, reaching the lowest temperature (same as in []). ð! 0! Þ Adiabatic change of the internal energy-level difference from E to E (see Fig. ), the internal temperature drops from to (e.g., adiabatic paramagnetic cooling). In contact with the cold reservoir, isothermal change of the internal energy difference to E (different from []). ð! 5Þ Adiabatic compression of the COM volume, increasing the COM temperature to. he internal temperature remains (same as in []). ð5! 6Þ Isochoric heating of the COM to the initial temperature. he internal temperature remains (same as in []). ð6! 6 0! Þ Adiabatic change of the internal energy-level difference from E to E, the internal temperature increases to. In contact with the hot reservoir, isothermal change of the internal energy difference to E (different from []).

6 66. Opatrný and M. O. Scully, Enhancing Otto-mobile Efficiency he efficiency of the cycle is h ¼ W ðottoþ þ W ðcarnotþ : ð0þ þ Using the expressions for the Otto cycle () and (7), taking into account that for the ideal gas C V ¼ k g ; ðþ and assuming the limiting case for the Carnot cycle, Eq. (9), one obtains the final formula (). As an example, let us assume a single-atom ideal gas with g ¼ 5=. Let us assume that the volume is compressed such that V =V ¼ =0, and the lowest temperature is ¼ =5. One finds for the Otto cycle that 0:7k ; ðþ W ðottoþ 0:50k ; ðþ h O 60:% : ðþ A two-level system Carnot cycle has in the limiting case 0:69k ; ð5þ W ðcarnotþ 0:55k ; ð6þ h C 80% : ð7þ he efficiency of the combined system is then, on using (0) h 69:7% ; ð8þ which is a substantial improvement of (). 5. Conclusion he improvement of the Otto cycle efficiency is based on running an additional thermodynamic cycle between the hottest and coldest temperatures, with working fluid represented by the internal degrees of freedom of the gas particles. In [] this additional cycle was realized by lasers and masers. he cycle of [] contained irreversible steps (e.g., an initially hot internal state at is being thermalized with a cold maser cavity at, etc.) which suggests that other improvements are possible. On the other hand, the cycle presented here is completely reversible. hus, given the temperatures, and, and the number N of internal states participating in the additional cycle, the formula () represents the maximum possible efficiency of the complete cycle. References [] H. E. D. Scovil and E. O. Schulz-DuBois, hree-level masers as heat engines, Phys. Rev. Lett. (959) 6. [] See for example E. Geva and R. Kosloff, On the classical limit of quantum thermodynamics in finite-time, J. Chem. Phys. 97 (99) 98;

7 Fortschr. Phys. 50 (00) S. Lloyd, Quantum-mechanical Maxwell s demon, Phys. Rev. A 56 (997) 7;. Feldmann and R. Kosloff, Performance of discrete heat engines and heat pumps in finite time, Phys. Rev. E6 (000) 77; R. Kosloff, E. Geva, and J. M. Gordon, Quantum refrigerators in quest of the absolute zero, J. Appl. Phys. 87 (000) 809 and references therein. [] M. O. Scully, he quantum afterburner, Phys. Rev. Lett., to be published. [] M. O. Scully, A. B. Matsko, N. Nayak, Y. V. Rostovtsev, and M. S. Zubairy, he quantum Otto-mobile: improving engine efficiency via lasing off exhaust gassess, in preparation. [5]. Numazawa, H. Kimura, M. Sato, and H. Maeda, Carnot magnetic refrigerator operating between.-k and 0-K, Cryogenics (99) 57; A. Bezaguet, J. Casascubillos, P. Lebrun, R. Losserandmadoux, M. Marquet, M. Schmidtricker, and P. Seyfert, Design and construction of a static magnetic refrigerator operating between.8-k and.5-k, ibid (99) 7, Suppl. ICEC. [6] Z. J. Yan and J. C. Chen, he characteristics of polytropic magnetic refrigeration cycles, J. Appl. Phys. 70 (99) 9; L. X. Chen and Z. J. Yan, Main characteristics of a Brayton refrigeration cycle of paramagnetic salt, ibid 75 (99) 9. [7] H. oftlund, A rotary Curie-point magnetic engine a simple demonstration of a Carnot-cycle device, Am. J. Phys. 55 (987) 8; G. Barnes, he cycles of the rotary Curie-point heat engine, ibid 57 (989) ; P. G. Mattocks, A 0.7-mW magnetic heat engine, ibid 58 (990) 55.

Chapter 12 Thermodynamics

Chapter 12 Thermodynamics 12.1 Thermodynamic Systems, States, and Processes System: definite quantity of matter with real or imaginary boundaries If heat transfer is impossible, the system is thermally