A short note on Reitlinger thermodynamic cycles

Transcription

1 short note on Reitlinger thermodynamic cycles melia arolina Saravigna eartment of lied Science and echnology, Politecnico di orino, orino, Italy bstract: It is well known that arnot cycle is the thermodynamic cycle which has the best thermal efficiency. However, an entire class of cycles exists that can have the same maximum efficiency: this class is that of the regenerative Reitlinger cycles. Here we discuss them. Keywords: hermodynamics, hermodynamic cycles, regenerative cycles, hermal efficiency. Generally, the arnot cycle is the only thermodynamic cycle that, during the lectures on hysics, is discussed as having the maximum thermal efficiency. his aroach yields the following result: it is less known that an entire class of cycles exists, having a cycle efficiency which is the same of the arnot cycle. his is the class of the regenerative Reitlinger cycles. hey consist of two isothermal and two olytroic rocesses of the same kind [,], so that the heat which is absorbed during a olytroic, is exactly the same that it is rejected on the other olytroic rocess. herefore, if we have a erfect regeneration of heat, by means of which the heat rejected during the olytroic is transferred to a thermal storage (the regenerator) and then transferred back to the working fluid, the thermal efficiency of the Reitlinger cycle equals that of the arnot cycle (in fact, it is a Reitlinger cycle too). Of all the Reitlinger cycles, the arnot cycle is unique in requiring the least regeneration, namely, none at all because its olytroics are adiabatics []. However, the mechanical work of the arnot cycle is not the best we can obtain between extremal states, as we can see from the diagram in Figure, which is comaring arnot and Stirling cycles, having the same temerature and volume extremes []. Figure : he figure (adated from Ref.) shows a arnot cycle inscribed in a Stirling cycle in a - diagram. he otimum cosntant buffer ressure is also shown. he work of the Stirling cycle is greated than the work of arnot cycle.

2 Let us note that the ideal Stirling cycle is also a Reitlinger cycle, having as olytroics two isochoric segments. It is the most oular examle of a cycle having the same thermodynamic efficiency of the arnot cycle; however, to attain this result, the Stirling cycle makes quite heavy demands on the rocess of regeneration [3]. Figure : he figure shows how a Reitlinger cycle can be different from a arnot cycle, in a - diagram. orking between the same isothermals, with the same thermal efficiency, a regenerative Reitlinger cycle can give more or less work, deending on the olytroic rocess the cycle is erforming between the same extremal states. s observed in [6], there are ten elementary ower cycles which follow from the combinations of five tyical thermodynamic changes of state (Fig. 3). In the Figure we can see them and the names of their inventors (for other cycles, see [7]). In [6], arnot, Ericsson and Stirling cycles are distinguished from the Reitlinger cycles, because they have a secific imortance in thermodynamics. hey are cycles containing the isothermal rocesses of comression and exansion, which have the most general form in the idealized cycles analyzed by Reitlinger [4,5]. s reviously told, in these cycles we have, besides the two isothermal rocesses, two olytroic regenerative rocesses. Figure 3: he elementary thermodynamic cycles (figure adated from [6]).

3 For any thermodynamic cycle, reversible or irreversible, after one cycle, the working fluid is again in its initial state and thus the change of its internal energy is zero. In this manner, the first rincile of thermodynamics tell us that the mechanical work roduced by the cycle is the difference of inut heat energy Qin minus the energy dissiated in waste heat Qout. Heat engines transform thermal energy into mechanical energy or work,, so that = Qin Qout. e can calculate the thermal efficiency of the cycle as the dimensionless erformance measure of the use of thermal energy. he thermal efficiency of a heat engine is the ercentage of heat energy that is transformed into work, so that: () Q in For a arnot engine, it is η = /H, where H, are the temeratures of the furnace and of the cold sink. Let us discuss the thermal efficiency of the Stirling cycle. Using a - diagram, the cycle aears as in the Figure 4. In the same figure, the Ericsson cycle and Reitlinger cycle are also shown. Figure 4: Stirling, Ericsson and Reitlinger cycles in - diagrams. he work can be easily calculated as: () nr In (), n is the number of moles and R the universal gas constant. Heat is gained by the thermodynamic system from the reversible isochoric transformation from to and during the isothermal ath. uring isochoric rocess, the heat gained is: Qisoc n. is the molar secific heat for an isochoric rocess. uring isothermal rocess, the heat gained is Q nr ( / ). 3 isot Let us note that, during the isochoric rocess, the fluid is obtaining heat from an infinite number of thermal reservoirs [8]. his same amount of heat is lost during the isochoric cooling rocess, with a thermal exchange with the same reservoirs. hen, for each of the infinite thermal reservoirs that we meet during the isochoric rocess, it haens what we see in the Figure 5. In this figure, we have two thermal machines that must have the same efficiency, to satisfy the second rincile of thermodynamics.

4 Figure 5: he two reversible cycles in the figure have the same efficiency. If it were not so, we should violate the second rincile of thermodynamics. Let us suose the efficiency of the right machine larger than that of the left one. Let us consider the same work roduced by the two machines, and oerate the machine on the left in reversed manner. It is easy to see that the net result of these two machines oerating together is that of transferring some heat from the low temerature reservoir to the high temerature reservoir, violating the lausius statement of the second rincile of thermodynamics. From an engineering ersective, the regenerator simly stores the heat and therefore the abovementioned thermal reservoirs are not involved. onsequently, considering the system made of working fluid and regenerator, the thermal efficiency is: (3) Q nr nr In (3), Q is the heat the system receives during the high temerature isothermal rocess. his efficiency is equal to that of a arnot cycle which is working between the same two isotermal rocesses. e can reeat the calculation for the Ericsson cycle. he work is: (4) n P nr nr ( ) nr nr nr nr n nr P ( ) nr nr nr In (4), is the molar secific heat at constant ressure. It is clear that the heat lost and gained during the two isobaric rocesses is the same. herefore, the thermal efficiency, in the case of a erfect regeneration, is given by: (5) Q nr nr 4

5 In (5), Q is the heat the system receives during the high temerature isothermal rocess. Let us conclude with a Reitling cycle, where olytroics are given by equations const and const. he molar secific heat of such olytroic rocess is α. Let us note that from olytroic equation we have (see Figure 4): (6) herefore, we have: (7) hen: (8) n nr ( ) nr nr n ( ) nr gain, we find a thermal efficiency of the system (fluid and regenerator), which is again equal to that of the arnot engine. herefore, since the olytroic index α can have any value, we have an infinite number of thermodynamic cycles that have the same value of thermal efficiency, equal to that of the arnot cycle when oerating between the same two isothermal rocesses. Let us stress that these cycles incororate a regenerative heat transfer rocess, in lace of adiabatic comression and exansion of the arnot cycle [5], or, if referred, an infinite number of monothermal rocesses, not influencing the efficiency of the cycle. Moreover, it is better to also remark that the fact of ossessing the same thermal efficiency does not mean that the same work is obtained from different reversible cycles, when they are oerating between the same extremal states. References [] J.R. Senft, Mechanical Efficiency of Heat Engines, ambridge University Press, 007. [] I. Kolin, he Evolution of the Heat Engine, Longman, 97. [3] J.R. Senft, n Introduction to Stirling Engines, Moriya Press, 993. [4] J. Reitlinger, Uber Kreisrozesse zwischen zwei isothermen. Z. Ost. Ing. rch. er [5] G. alker, ryocoolers, Part : Fundamentals, Plenum Press 983. [6] I. Kolin, S. Koscak-Kolin, M. Golub, Geothermal Electricity Production by means of the Low emerature ifference Stirling Engine, Proceedings orld Geothermal ongress 000, Kyushu - ohoku, Jaan, May 8 - June 0, 000, [7] J.Selwin Rajadurai, hermodynamics and hermal Engineering, New ge International, 003. [8] P. Mazzoldi, M. Nigro,. oci, Fisica, S.E.S