Some considerations about thermodynamic cycles

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1 Home Seach Collections Jounals About Contact us My IOPscience Some consideations about themodynamic cycles This aticle has been downloaded fom IOPscience. Please scoll down to see the full text aticle. 0 Eu. J. Phys View the table of contents fo this issue, o go to the jounal homepage fo moe Download details: IP Addess: The aticle was downloaded on //0 at 3:5 Please note that tems and conditions apply.

2 IOP PUBLISHING Eu. J. Phys EUROPEAN JOURNAL OF PHYSICS doi:0.088/ /33//00 Some consideations about themodynamic cycles M F Feeia da Silva Depatamento de Física, Univesidade da Beia Inteio, Rua Maquês d Ávila e Bolama, Covilhã, Potugal mffs@ubi.pt Received 8 Septembe 0, in final fom 30 Septembe 0 Published 3 Octobe 0 Online at stacks.iop.og/ejp/33/3 Abstact Afte completing thei intoductoy studies on themodynamics at the univesity level, typically in a second-yea univesity couse, most students show a numbe of misconceptions. In this wok, we identify some of those eoneous ideas and ty to explain thei oigins. We also give a suggestion to attack the poblem though a systematic and detailed study of vaious themodynamic cycles. In the meantime, we deive some useful elations.. Intoduction In univesity physics textbooks at the intoductoy level [ 3], the analysis of heat engines and heat pumps/efigeatos is typically included in the study of the second law of themodynamics. Heat engines ae usually used to intoduce the Kelvin Planck statement of the second law; heat pumps/efigeatos ae used to intoduce the Clausius statement of the second law. The schematic diagams of these machines, based on the consevation of enegy fist law of themodynamics and the exchange of heat and wok, allow us to explain, in a satisfactoy manne, the concepts of efficiency of a heat engine, coefficient of pefomance COP of a heat pump and coefficient of pefomance of a efigeato. In paticula, the Canot cycle is discussed in detail, and the Canot theoem, which links this cycle with heat engines of geatest efficiency and heat pumps/efigeatos of geatest coefficients of pefomance, is poved. The student also woks out a numbe of othe cycles Otto cycle, Diesel cycle, etc while studying heat engines, sometimes diectly in the text [], sometimes though poposed poblems [, 3]. Nevetheless, those cycles do not appea duing the study of heat pumps/efigeatos. At the same time, and gadually thoughout this study, the concept of entopy is intoduced, and the diffeence between evesible and ievesible pocesses is explained. My teaching expeience has shown that the student concludes the study of these topics with some impotant misconceptions, namely // $33.00 c 0 IOP Publishing Ltd Pinted in the UK & the USA 3

3 4 MFFdaSilva Eoneous compehension of the idea of cycle invesion. Difficulty in undestanding the impotance of the Canot cycle; specifically, difficulty in undestanding the tue meaning of the sentence Canot cycle is evesible, while othe cycles ae ievesible. The student fequently thinks in this way: If, afte each cycle, the system etuns to its initial state, and the two esevois have the same oiginal tempeatues, whee can be the ievesibility? O in this way: If a cycle could be inveted, then it is automatically evesible! Eoneous belief in that the unique schematic diagams allowed by the two laws of themodynamics ae the diagam coesponding to a heat engine and the diagam coesponding to a heat pump/efigeato. Pofound belief that it is enough to invet the cycle of any heat engine to automatically tansfom it into the coesponding heat pump/efigeato. The fist misconception has a clealy identifiable souce: only the Canot cycle is explicitly inveted in textbooks. This difficulty is thus elatively easy to solve, by showing othe examples of invesion to the student. The second misconception has anothe oigin: the student aely calculates entopy changes while dealing with heat engines in specific cycles; the Canot cycle is, one moe time, the only exception. So the confusion between the entopy of the system and the entopy of the univese is vey usual in this context. The last two misconceptions diectly esult fom the lack of vaious examples of invesion of themodynamic cycles in most texts. This wok has a pimay goal to show how these misconceptions can be solved. We will poceed to a systematic and caeful analysis of vaious specific cycles, all well known in the study of heat engines; natually, we will begin with the Canot cycle in ode to show its unique popety. In all cases, we will study the cycle of the heat engine, and then we will poceed to its invesion. Wheneve possible we will get elations between the elevant paametes. We will only conside the simplest vesions of those cycles, that is, the ideal vesions. Hee and thee we will efe to some engineeing applications, but only as seconday infomation. We will also ignoe all pactical difficulties in the implementation of specific cycles as well as the coesponding solutions. Finally, little histoical digessions will be included as footnotes, fo the sake of completeness.. Theoy and notation We wite the fist law of themodynamics in the fom U Q + W, whee U epesents the change of intenal enegy of the system, Q is the heat absobed by the system and W is the wok done on the system. We will focus ou attention to hydostatic systems systems descibed by themodynamic vaiables V volume, p pessue and T tempeatue following cycles, so U 0, Q T ds, W p dv ; hee, S epesents the entopy of the system. We admit the existence of only two themal esevois hot body and cold body, at diffeent but constant tempeatues, in contact with the system; the fist law of themodynamics assumes the fom 0 Q cycle + W cycle Q h + Q c + W cycle 0, 3

4 Some consideations about themodynamic cycles 5 Resevoi at T h Resevoi at T h Q h Q h System System W cycle W cycle Q c Q c Resevoi at T c Resevoi at T c a b Figue. Schematic diagam of a a heat engine and b a heat pump/efigeato. whee Q h epesents the heat absobed by the system, duing the cycle, fom the esevoi at highe tempeatue labelled T h and Q c epesents the heat absobed by the system, duing the cycle, fom the esevoi at lowe tempeatue labelled T c. We will designate as a heat engine any machine woking in cycles that allows us to extact wok fom the two esevois. The schematic epesentation of a heat engine is shown in figue a. Since in this case W cycle < 0, Q h > 0 and Q c < 0, we have Q h Q c W cycle 0 W cycle Q h Q c. 4 The efficiency of the heat engine is given by ɛ W cycle Q h Q c Q c <. 5 Q h Q h Q h Since W cycle < 0 and Q cycle > 0 in a heat engine, its cycle, when epesented in a volume pessue V p diagam o in an entopy tempeatue S T diagam, must be pefomed in a clockwise diection, on account of elations. By inveting all aows of figue a, we get the taditional schematic epesentation of a heat pump/efigeato, shown in figue b. Since in this case W cycle > 0, Q h < 0 and Q c > 0, we have Q h + Q c + W cycle 0 W cycle Q h Q c. 6 The COPs of the heat pump HP and the efigeato R ae Q h W cycle Q h Q h Q c >, 7 COP R Q c W cycle Q c Q h Q c, 8

5 6 MFFdaSilva Pessue adiabatics 3 isothem T 4 isothem T V V Volume Figue. Volume pessue diagam of the Canot cycle. and it is clea that COP R. 9 Since W cycle > 0 and Q cycle < 0 in a heat pump/efigeato, its cycle, when epesented in av p diagam o in an S T diagam, must be ealized in a counteclockwise diection, on account of elations. In any case, the change of the entopy of the univese afte one cycle is S univese S system + S esevoi at T h + S esevoi at T c Qh T h 0+ Q h T h + Q c T c + Q c. 0 T c We will assume that the system o woking substance is always an ideal gas, and that phase changes do not occu. We will epesent the numbe of moles by n, the gas constant by R, the mola heat capacities at constant volume and at constant pessue by c V and c p, espectively, and the adiabatic coefficient by γ. Some elations between these quantities ae pv nrt, c p c V R, γ c p >, c V R c V, c p γr. 3. The Canot cycle Let us conside the Canot cycle shown in figue, with two isothemal pocesses and two adiabatic pocesses, whee >and T <T. We define T T T > 0. Since each adiabatic cuve satisfies the equation TV γ constant, we can wite { T V γ T V γ T 3 V γ 3 T 4 V γ 4 V3 V γ V4 Intoduced by Nicolas Léonad Sadi Canot in 84. V γ V 3 V V 4 V,

6 Some consideations about themodynamic cycles 7 because T 3 T and T 4 T. The paamete is called compession atio and elates the initial and final volumes of the gas duing isothemal pocesses. Let us suppose that the cycle is done in the clockwise diection. In pocesses and 3 4, the gas is themally isolated, in pocess 3, the gas is in contact with a themal esevoi at tempeatue T h T, and in pocess 4, the gas is in contact with a themal esevoi at tempeatue T c T. We have V3 Q 0 ; Q 3 W 3 nrt ln nrt ln ; V V4 Q 34 0 ; Q 4 W 4 nrt ln nrt ln, so V Q h Q 3 nrt ln >0, Q c Q 4 nrt ln <0, Q cycle Q h + Q c nr T ln >0, W cycle Q cycle nr T ln <0. The gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; this behaviou defines a heat engine. Accoding to 5, the efficiency of the Canot engine is given by ɛ Canot cycle Q c Q h nrt ln nrt ln T T 3 T T and it depends only on the tempeatues of the two esevois. Using 0, the entopy change of the univese duing a cycle of the Canot engine is SCanot univese cycle, Q h Q c nrt ln + nrt ln 0, 4 T h T c T T that is, the clockwise Canot cycle is evesible. This popety makes the Canot engine a vey special engine: it is the unique evesible heat engine functioning with only two esevois. Any othe cycle could be pefomed in a evesible way, but that would equie a vitually infinite numbe of themal esevois. Let us ty to tansfom this heat engine into a heat pump/efigeato. In ode to do so, we invet the cycle to the counteclockwise diection. In pocess 4, the gas is in contact with a esevoi at tempeatue T c T c T, in pocesses 4 3 and, the gas is themally isolated, and in pocess 3, the gas is in contact with a esevoi at tempeatue T h T h T. The pimed quantities efe to the counteclockwise cycle. We will adopt this notation fom now on. We have Q 4 Q 4 nrt ln ; Q 43 0 ; Q 3 Q 3 nrt ln ; Q 0 ; so Q h Q 3 nrt ln <0, Q c Q 4 nrt ln >0, Q cycle Q h + Q c nr T ln <0, W cycle Q cycle nr T ln >0. Natually, Q cycle Q cycle and W cycle W cycle. Now the gas is ejecting heat to the esevoi at highe tempeatue, is absobing heat fom the esevoi at lowe tempeatue, and it is doing wok on the gas; this behaviou defines a heat pump/efigeato. In shot, we see Conventionally, we think of a clockwise cycle, epesenting the heat engine. As the name indicates, the compession atio elates the initial and final volumes duing the isothemal compession pocess, namely the volumes V 4 and V. The expansion atio is defined as the atio between the final and initial volumes duing the isothemal expansion pocess; in this case V 3 V. We can also define the pessue atio as the atio between the final and initial pessues duing the isothemal compession pocess; hee, p p p. Since p 4 V p 4 V 4 and p V p 3 V 3,wehave p p p 4 V 4 V and p p 3 V 3 V, on account of. Thus in the Canot cycle, the compession atio,the expansion atio and the pessue atio p ae the same.

7 8 MFFdaSilva that the staightfowad invesion of the cycle of a Canot engine automatically geneates a Canot heat pump/efigeato. Accoding to 7 and 8, the COP of the Canot heat pump and the Canot efigeato ae Canot cycle Q h W cycle nrt ln nr T ln T T, 5 COP R Canot cycle Q c W cycle nrt ln nr T ln T T. 6 We can veify the geneal elation 9, and we also obseve that Canot cycle. 7 ɛ Canot cycle Finally, the change in the entopy of the univese duing a cycle of this Canot heat pump/efigeato is given by SCanot univese cycle, Q h Q c nrt ln nrt ln 0, 8 T h T c T T which shows that the counteclockwise Canot cycle is also evesible; this popety makes the Canot heat pump/efigeato a vey special heat pump/efigeato. 4. Stiling cycle Let us conside the Stiling cycle 3 shown in figue 3. This cycle has a clea esemblance to the Canot cycle 4 ; it has two isothemal pocesses and two isochoic pocesses eplacing the adiabatic pocesses, whee > and T < T. We define T T T > 0, as befoe. Let us assume that the cycle opeates in the clockwise diection. In pocesses 3, the gas is in contact with a themal esevoi at tempeatue T h T, and in pocesses 3 4, the gas is in contact with a themal esevoi at tempeatue T c T.Wehave Q nc V T n R T ; Q V3 3 W 3 nrt ln nrt ln ; V Q 34 nc V T 34 n R T ; Q V4 4 W 4 nrt ln nrt ln, so V Q h Q + Q 3 nr T ln + T > 0, Q c Q 34 + Q 4 nr T ln + T < 0, Q cycle Q h + Q c nr T ln >0, W cycle Q cycle nr T ln <0. Note that the values of Q cycle and W cycle in the Stiling cycle ae the same as the coesponding values in the Canot cycle. Since Q h > 0, Q c < 0 and W cycle < 0, the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; these popeties chaacteize a heat engine. 3 Used by Robet Stiling in 86 in his ai engine; that heat engine was impoved in 840 by Robet s bothe, James Stiling In this cycle the compession atio, the expansion atio and the pessue atio also coincide. The equality of the fist two atios is automatic, and the equality with the thid atio is poved as in the pevious section.

8 Some consideations about themodynamic cycles 9 Pessue 3 isothem T 4 isothem T V V Volume Figue 3. Volume pessue diagam of the Stiling cycle. The efficiency of the Stiling engine is given by ɛ Stiling cycle W cycle Q h + nr T ln nr T ln + T γ T T T T γ ln T ln [ + T ln T γ T ln ɛ Canot cycle + ɛ, 9 Canot cycle γ ln whee we have used elation 3 fo the efficiency of the Canot engine. This expession clealy shows that ɛ Stiling cycle <ɛ Canot cycle and that ɛ Stiling cycle ɛ Canot cycle if. It can also be witten in the fom ɛ Stiling cycle ɛ Canot cycle ln. 0 The entopy change of the univese duing a cycle of the Stiling engine is SStiling univese cycle, Q h Q c nr T ln + T γ + nr T ln + T γ T h T c T T nr T nr T > 0, T T T T that is, the clockwise Stiling cycle is ievesible. Let us ty to tansfom this heat engine into a heat pump/efigeato, as in the pevious section. In ode to do so, we invet the cycle to the counteclockwise diection. We must be caeful hee when identifying the esevois that ae in contact with the gas in the vaious pocesses. In pocesses 4 and, the gas should be in contact with a esevoi at tempeatue T c T c T, and in pocesses 4 3, the gas should be in contact with a esevoi at tempeatue T h T h T. Then we have Q 4 Q 4 nrt ln ; Q 43 Q 34 n R T ; ]

9 0 MFFdaSilva Resevoi at T h Resevoi at T h Q h Q h System System W cycle W cycle Q c Q c Resevoi at T c Resevoi at T c a b Figue 4. Schematic diagams of situations a and b descibed in sections 4 and 5. so T, Q c Q 4 + Q nr Q 3 Q 3 nrt ln ; Q Q n R T, Q h Q 43 + Q 3 nr T ln T ln Q cycle Q h + Q c nr T ln <0, W cycle Q cycle nr T ln >0. T, It is vey impotant to compae Q h and Q c with Q h and Q c, espectively, and undestand the oigin of the diffeence. The next obsevation to do is this: although wok is being done on the gas, thee is no waanty that Q h < 0 condition chaacteizing a heat pump o Q c > 0 condition chaacteizing a efigeato. Actually, we have thee possible situations. a T ln T < 0 ln < T γ γ T <exp [ ] T γ T. In this case, Q h > 0 and Q c < 0, that is, the gas is absobing heat fom the esevoi at highe tempeatue and ejects heat to the esevoi at lowe tempeatue. This situation is epesented by the diagam shown in figue 4a, and it does not coespond to a heat engine, no a heat pump/efigeato; it is descibed by the elation W cycle Q c Q h. b T ln T γ > 0 and T ln T exp [ ] [ ] T γ T <<exp T γ T. In this case, Q h < 0 and Q c γ < 0 T γ T < ln < T γ T < 0, that is, the gas ejects heat to the esevoi at highe tempeatue and to the esevoi at lowe tempeatue. This situation is epesented by the diagam shown in figue 4b, and it does not coespond to a heat engine, no a efigeato; it is descibed by the elation W cycle Q c + Q h. We could say that it

10 Some consideations about themodynamic cycles achieves the same goal as that of a heat pump because it sends heat to the esevoi at highe tempeatue; nevetheless, its coefficient of pefomance is low because Stiling cycle,b Q h nr T ln T W cycle γ T nr T ln T ln, and we see that 0 < Stiling cycle, b <. c T ln T > 0 ln > T γ γ T >exp [ ] T γ T. In this case, Q h < 0 and Q c > 0, that is, the gas absobs heat fom the esevoi at lowe tempeatue, and ejects heat to the esevoi at highe tempeatue; these popeties define a heat pump/efigeato. The coesponding coefficients of pefomance ae Stiling cycle, c Q h W cycle nr T ln T γ T nr T ln T ln, T ln T γ COP R Stiling cycle, c Q c W cycle nr nr T ln with < Stiling cycle, c < T T ln, 3 Canot cycle and 0 < COP R Stiling cycle, c < COP R Canot cycle. These two coefficients veify the geneal elation 9, and using equations 3 and 0, we obtain Stiling cycle, c ɛ Canot cycle ɛ Stiling cycle ɛ Canot cycle ; 4 ɛ Canot cycle ɛ Stiling cycle a simila elation is also satisfied by the Canot heat pump, as is easy to pove. Thus, we can say that the invesion of the cycle of the Stiling engine does not necessaily epesent a heat pump/efigeato; it only occus if cetain conditions ae satisfied. The change in the entopy of the univese duing the inveted counteclockwise Stiling cycle is S univese Stiling cycle, Q h T h Q c T c nr T nr T ln T γ T T T nr nr T ln T γ T T T T > 0 ; 5 this esult confims the ievesibility of this cycle and coincides with the esult obtained fo the clockwise cycle. It is woth mentioning that conditions descibing the peviously efeed situations a, b and c can be witten in an altenate fom; if we look at the expessions of W cycle and S univese Stiling cycle that is,, we see that, fo situation a, ln < T nr T ln < nr T, T T W cycle <T S univese Stiling cycle situation a.

11 MFFdaSilva Engine Heat Pump Heat Pump / Refigeato 0 T ΔS univese Stiling cycle T ΔS univese Stiling cycle W cycle Figue 5. Behaviou of the Stiling cycle accoding to W cycle values. Pessue 3 4 isothem T isothem T V V Volume Figue 6. Volume pessue diagam of the Eicsson cycle. In a simila way, we obtain T SStiling univese cycle <W cycle <T SStiling univese cycle situation b. T SStiling univese cycle <W cycle situation c. Thus, we can intepet T SStiling univese cycle as the minimum wok we have to do on the gas to implement a Stiling heat pump, and T SStiling univese cycle as the minimum wok we have to do on the gas to implement a Stiling efigeato. Figue 5 summaizes all these conclusions. 5. Eicsson cycle Let us now conside the Eicsson cycle 5, shown in figue 6. This cycle also shows stong similaities with the Canot cycle 6, and has two isothemal pocesses and two isobaic pocesses which eplace the adiabatic pocesses, whee > and T <T.Let T T T > 0. Since isothemal pocesses ae descibed by the equation pv constant, we have { p V p 4 V 4 p p 4 V 4 V p V p 3 V 3 p p 3 V V 3 V V V V 5 Used by John Eicsson in his extenal combustion engine in Once moe, the compession atio, the expansion atio and the pessue atio ae equal in this cycle, as is clea in 6.

12 Some consideations about themodynamic cycles 3 because p p and p 3 p 4. Let us assume that the cycle occus in the clockwise diection. In pocesses 3, the gas is in contact with a themal esevoi at tempeatue T h T, and in pocesses 3 4, the gas is in contact with a themal esevoi at tempeatue T c T. We have Q nc p T n γr T ; Q V3 3 W 3 nrt ln nrt ln ; V Q 34 nc p T 34 n γr T ; Q V4 4 W 4 nrt ln nrt ln, so V Q h Q + Q 3 nr T ln + γ T > 0, Q c Q 34 + Q 4 nr T ln + γ T < 0, Q cycle Q h + Q c nr T ln >0, W cycle Q cycle nr T ln <0. Note that the values of Q cycle and W cycle in the Eicsson cycle ae the same values calculated in the Canot and Stiling cycles. Since Q h > 0, Q c < 0 and W cycle < 0, the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue, and is doing wok, so this cycle behaves as a heat engine. The efficiency of the Eicsson engine is given by ɛ Eicsson cycle W cycle Q h T T + γ T T γ ln nr T ln nr T ln + γ T γ T ln T ln [ + γ T γ T ln ɛ Canot cycle + γɛ, 7 Canot cycle γ ln whee we have used elation 3 fo the efficiency of the Canot engine. Compaing this expession with 9, we see that 0 <ɛ Eicsson cycle <ɛ Stiling cycle <ɛ Canot cycle < and also that ɛ Eicsson cycle ɛ Canot cycle if. This can also be witten as γ ɛ Eicsson cycle ɛ Canot cycle ln. 8 The entopy change of the univese duing a cycle of the Eicsson engine is SEicsson univese cycle, Q h Q c nr T ln + γ T γ + nr T ln + γ T γ T h T c T T nrγ T T T nrγ ] T T T > 0, 9 so the clockwise Eicsson cycle is ievesible. Let us ty, once again, to tansfom this heat engine into a heat pump/efigeato. In ode to do so, we invet the cycle to the counteclockwise diection and caefully identify the esevois that ae in contact with the gas in the vaious pocesses. In pocesses 4 and the gas is in contact with a esevoi at tempeatue T c T c T, and in pocesses 4 3 the gas is in contact with a esevoi at tempeatue T h T h T.So Q 4 Q 4 nrt ln ; Q 43 Q 34 n γr T ;

13 4 MFFdaSilva Q 3 Q 3 nrt ln ; Q h Q 43 + Q 3 nr Q Q n γr T, T ln γ T so, Q c Q 4 + Q nr T ln γ T, Q cycle Q h + Q c nr T ln <0, W cycle Q cycle nr T ln >0. Once moe, it is instuctive to compae Q h and Q c with Q h and Q c, espectively. As in the pevious section, wok is done on the gas but thee is no waanty that Q h < 0 the main condition epesenting a heat pump o Q c > 0 the main condition epesenting a efigeato. We have thee possible scenaios: a <exp [ γ T γ T ] ;bexp [ γ T γ T ] <<exp [ γ T γ T ] ;c>exp [ γ T γ T ]. Scenaio a is shown in figue 4a and does not coespond to a heat pump, no to a efigeato. Scenaio b is shown in figue 4b; we can say that it behaves as a heat pump with a low coefficient of pefomance, given by Eicsson cycle, b Q h W cycle nr and we can wite 0 < T ln γ T γ nr T ln T T γ ln, Eicsson cycle, b < COPHP Stiling cycle, b <. Scenaio c is unique that coesponds to a heat pump/efigeato, with the coefficients of pefomance Eicsson cycle, c Q h W cycle nr T ln γ T γ T nr T ln T γ ln, 30 COP R Eicsson cycle, c Q c nr T ln γ T W cycle γ nr T ln T T γ ln. 3 These satisfy elations < Eicsson cycle, c < Stiling cycle,c < Canot cycle and 0 < COP R Eicsson cycle, c < COPR Stiling cycle,c < COPR Canot cycle, and veify the geneal elation 9 and still anothe elation simila to 4: Eicsson cycle, c ɛ Canot cycle ɛ Eicsson cycle ɛ Canot cycle. 3 ɛ Canot cycle ɛ Eicsson cycle Thus, the invesion of the cycle does not necessaily epesent a heat pump/efigeato. The entopy change of the univese afte one counteclockwise Eicsson cycle is SEicsson univese cycle, Q h Q c nr T ln γ T γ nr T ln γ T γ T h T c T T nrγ T T T nrγ T T T > 0 ; 33 this is the same esult as that of the clockwise cycle. Finally, and in a simila way as we did fo the Stiling cycle, pevious scenaios a c can be witten as W cycle <T SEicsson univese cycle scenaio a,

14 Some consideations about themodynamic cycles 5 T S univese Eicsson cycle <W cycle <T S univese Eicsson cycle scenaio b, T SEicsson univese cycle <W cycle scenaio c, so we can intepet T SEicsson univese cycle as the minimum wok we have to do on the gas to get an Eicsson heat pump, and T SEicsson univese cycle as the minimum wok we have to do on the gas to get an Eicsson efigeato. A diagam simila to figue 5 would show these esults. It is woth obseving that Q + Q 34 0 in both Stiling and Eicsson cycles, that is, the heats exchanged by the system duing the non-isothemal pocesses ae given by opposite numbes. This fact has been exploed in engineeing though the concept of egeneation: we include in the system a device called the egeneato which stoes the heat ejected by the gas duing pocess 3 4 and tansfes it back to the gas in pocess. It is easy to pove that the Stiling and Eicsson cycles with egeneation have the same efficiency and the same coefficients of pefomance as the Canot cycle. The eade inteested in this topic could find it in most engineeing themodynamics books [4]. In this section, and in the pevious section, we have consideed cycles that maintain the two isothemal pocesses of the Canot cycle and eplace the two adiabatic pocesses by isobaic and isochoic pocesses, espectively. In the following sections, we will maintain the adiabatic pocesses of the Canot cycle and eplace the isothemal pocesses. 6. Otto cycle Let us conside the Otto cycle 7 shown in figue 7, with two adiabatic pocesses and two isochoic pocesses which eplace the isothemal pocesses in the Canot cycle, whee > is the compession atio. Obseve that hee the paamete elates the exteme volumes of the gas along the adiabatic pocesses instead of along the isothemal pocesses 8. The points of the cycle with the highest and lowest tempeatues ae also shown. We define T T 3 T > 0. The eason fo choosing T 3 T instead of T 3 T will be clea soon. Since one of the equations of adiabatic cuves is TV γ constant, we can wite { T V γ T V γ T 3 V γ 3 T 4 V γ 4 T T T 3 T 4 V V V4 V 3 γ γ γ γ T T 3 T T 4 <. 34 Let us assume that the cycle is ealized in the clockwise diection. In pocesses and 3 4, the gas is themally isolated; in pocess 3, the gas is in contact with a themal esevoi at tempeatue T h T 3, and in pocess 4, the gas is in contact with a themal esevoi at tempeatue T c T. We have Q 0 ; Q 3 nc V T 3 n R T 3 T nr T ; Q 34 0 ; Q 4 nc V T 4 n R T 4 T nrt 4 T T 4 nrt 4 T T 3 7 In honou of Nikolaus August Otto 83 9, who designed an intenal combustion engine with this cycle in 86. The Otto cycle is also known as the Beau de Rochas cycle because Alphonse Beau de Rochas actually was the fist to patent a fou-cycle engine using this cycle, in The same obsevation applies to the othe paametes, and p. In the Otto cycle the expansion atio and the compession atio ae automatically equal, but the pessue atio p is highe: p p p p 3 p 4 γ >;this elation is easily poved by using the equation pv γ constant of adiabatic cuves.

15 6 MFFdaSilva Pessue adiabatics T max 3 4 T min Figue 7. Volume pessue diagam of the Otto cycle. V V Volume Q h Q 3 nr T > 0, Q cycle Q h + Q c nr T W cycle Q cycle nr T nrt 4 T 3 T nr T T 4 T 3 Q c Q 4 nr T γ γ > 0, < 0. nr T T 3 < 0, γ γ, Thus, the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; this defines a heat engine. The efficiency of the Otto engine is given by ɛ Otto cycle W cycle Q h nr T γ γ nr T γ so, 35 γ and it depends only on the compession atio and on the natue of the gas. Since T and T 3 ae the tempeatues of the two esevois, it is clea that ɛ Otto cycle <ɛ Canot cycle because ɛ Canot cycle T T 3, whilst ɛ Otto cycle T 4 T 3, and T 4 >T. The entopy change of the univese afte one cycle of the Otto engine is SOtto univese cycle, Q h Q c T h nr T T 3 nr T γ + T c T 3 T3 T nr T γ T 4 T 3 T4 T nr T T T 3 nr T > 0, 36 T T 3 so the clockwise Otto cycle is also ievesible. It should be noted that, opposite to pevious cycles, T does not epesent hee the tempeatue diffeence between the two esevois, but the ange of tempeatues ove which the ievesible heating pocess occus. This new

16 Some consideations about themodynamic cycles 7 intepetation can be extended to pevious cycles. Also note that SOtto univese cycle, can be witten in the fom SOtto univese cycle, nr T4 T T3 T nr Tc Th, 37 T T 3 T c T h whee T c T 4 T T T 4 is the ange of tempeatues ove which the ievesible cooling pocess occus, and T h T 3 T T 3 T is the ange of tempeatues ove which the ievesible heating pocess occus. Let us ty to tansfom this heat engine into a heat pump/efigeato. In ode to do so, we invet the cycle to the counteclockwise diection and caefully identify the esevois that ae in contact with the gas in the vaious pocesses. In this case, we have an inteesting situation: thee is no need to employ the same themal esevois used in the clockwise cycle 9 tempeatues T 3 and T ; we can use esevois at tempeatues T 4 and T. Thus, in pocess 4, the gas is in contact with a esevoi at tempeatue T 4 ; in pocesses 4 3 and, the gas is themally isolated; and in pocess 3, the gas is in contact with a esevoi at tempeatue T.Wehave Q 4 Q 4 nr T ; Q γ 43 0 ; Q 3 Q 3 nr T ; Q 0. Since Q 4 > 0 and Q 3 < 0, if we want this cycle to epesent a heat pump/efigeato we have to be sue that Q 4 Q c and Q 3 Q h, that is, T 4 should be the tempeatue of the esevoi at lowe tempeatue, and T must be the tempeatue of the esevoi at highe tempeatue; that is possible only if T 4 <T it should be noted, looking at figue 7, that this condition is not guaanteed apioi. Thus we additionally assume that T c T 4 <T T h and we will have Q h Q 3 nr T < 0, Q c Q 4 nr T > 0, γ Q cycle Q h + Q c nr T < 0, γ W cycle Q cycle nr T > 0. γ The coefficients of pefomance of this inveted Otto cycle ae Otto cycle Q h W cycle COP R Otto cycle Q c W cycle nr T γ nr T γ γ nr T γ γ nr T γ γ These two coefficients veify the geneal elation 9 and >, 38 γ γ γ > Otto cycle. 40 ɛ Otto cycle The entopy change of the univese duing the counteclockwise Otto cycle is S univese Otto cycle, Q h T h nr Q c T c nr T γ T nr T T 4 γ T 3 T 4 nr T T T 3 T T T 3 > 0, 4 9 Actually, if we ty to use the same esevois we will get a situation of the type shown in figue 4a, which does not epesent a heat engine, no a heat pump/efigeato. We leave the details to the eade.

17 8 MFFdaSilva Pessue adiabatics T max 3 4 T min V V Volume Figue 8. Volume pessue diagam of the Joule cycle. the same esult obtained fo the clockwise cycle, so we can wite it in the fom 37. Finally, compaing the expessions fo W cycle and Sunivese Otto cycle, we can eadily show that condition T 4 <T is equivalent to condition W cycle >T SOtto univese cycle ; this allows us, as in pevious sections, to intepet the value of T SOtto univese cycle as the minimum wok that we have to do on the gas to implement an Otto heat pump/efigeato. Note that T epesents the tempeatue T h of the hot esevoi used in the heat pump/efigeato. 7. Joule cycle Let us conside the Joule cycle shown in figue 8, with two adiabatic pocesses and two isobaic pocesses which substitute the isothemal pocesses in the Canot cycle, whee >isthe compession atio 0. Depending on the context, this cycle is also known as the Bayton cycle, Stoddad cycle, Rankine cycle o Bell Coleman cycle. The points of the cycle with the highest and lowest tempeatues ae also shown. We define, as befoe, T T 3 T > 0. 0 In the Joule cycle, as in the Otto cycle, the expansion and compession atios have the same value andthe pessue atio p γ has a highe value, as we can see in 4. The designations Joule cycle and Bayton cycle have histoical eason: although the cycle was used by Eicsson in 833, it was not successful at the time; James Pescott Joule poposed it in 85 and Geoge Bayton was the fist to implement it with success in 87. In both cases, the cycle is elated to a heat engine whose woking fluid is a gas, with no phase changes e.g. gas tubine. The name Stoddad cycle is in honou of Elliott Joseph Stoddad 859?, who used it in his 99 and 933 extenal combustion engines. The designation Rankine cycle, in honou of William John Macquon Rankine 80 7, applies when the woking fluid used in the heat engine typically steam suffes a phase change duing the cycle. The name Bell Coleman cycle efes to the counteclockwise efigeation cycle used by Heny Bell , John Bell and Joseph James Coleman

18 Some consideations about themodynamic cycles 9 One equation fo adiabatics is pv γ constant; then, since p p 3 and p p 4,we have p γ { V p V γ p V γ γ p p 3 V γ 3 p 4V γ V 4 p γ V 4 V. 4 3 V4 V 3 V p 4 V 3 Adiabatics ae also descibed by TV γ constant o T γ p γ constant; then γ T V γ T V γ T T V V γ { T T γ p γ T γ p γ γ 3 γ T γ. 43 T γ 3 p γ 3 T γ T 4 p γ T 3 T T 4 T T 3 T T 4 T T 4 4 Let us assume that the cycle opeates in the clockwise diection. In pocesses and 3 4, the gas is themally isolated; in pocess 3, the gas is in contact with a themal esevoi at tempeatue T h T 3 ; and in pocess 4, the gas is in contact with a themal esevoi at tempeatue T c T.Wehave Q 0 ; Q 3 nc p T 3 n R T 3 T nr T γ ; Q 34 0 ; Q 4 nc p T 4 n γr T 4 T nrγ T 4 nrγ T 4 T 3 T nr T γ T 3 Q h Q 3 nr T γ > 0, Q c Q 4 nr T γ Q cycle Q h + Q c nr T γ W cycle Q cycle nr T γ γ γ > 0, < 0. T T 4 T 4 nr T γ T 3 nrγ T 4, γ γ < 0, T T 3 Thus, the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; thus we have a heat engine. The efficiency of the Joule engine is given by ɛ Joule cycle W cycle Q h nr T γ γ γ nr T γ γ so, 44 γ and is equal to the efficiency of the Otto engine; it depends only on the compession atio and on the natue of the gas. The entopy change of the univese afte one cycle of the Joule engine is SJoule univese cycle, Q h Q nr T γ c γ + T h T c T 3 nr T γ γ T T T nr T γ T T 3 nrγ T > 0, 45 T T 3 so the clockwise Joule cycle is also ievesible. Sometimes this efficiency is expessed as a function of the pessue atio p ; since p /γ, we can wite ɛ Joule cycle γ /γ. p

19 30 MFFdaSilva As in the pevious section, T does not epesent the tempeatue diffeence between the two esevois, but the ange of tempeatues ove which the ievesible heating pocess occus. And it is also possible to ewite this entopy change in the fom SJoule univese cycle, nrγ Tc Th. 46 T c T h In ode to tansfom this heat engine into a heat pump/efigeato, we poceed as we did in the Otto cycle: we invet the cycle, ealizing it in the counteclockwise diection, and we use two new esevois at tempeatues T 4 and T. In pocess 4, the gas is in contact with a esevoi at tempeatue T 4 ; in pocesses 4 3 and, the gas is themally isolated; in pocess 3, the gas is in contact with a esevoi at tempeatue T.Wehave Q 4 Q 4 nr T γ ; Q γ 43 0 ; Q 3 Q 3 nr T γ Imposing now the condition T c T 4 <T T, we will have h Q h Q 3 nr T γ < 0, Q c Q 4 nr T γ > 0, γ Q cycle Q h + Q c nr T γ < 0, γ W cycle Q cycle nr T γ γ > 0. The coefficients of pefomance of this inveted Joule cycle ae 3 Joule cycle Q h W cycle COP R Joule cycle Q c W cycle nr T γ γ nr T γ γ γ nr T γ γ γ nr T γ γ γ ; Q 0. >, 47 γ γ γ > These two coefficients veify the geneal elation 9 and Joule cycle. 49 ɛ Joule cycle The entopy change of the univese afte one counteclockwise Joule cycle is SJoule univese cycle, Q h T h Q nr T γ nr T γ T 4 c γ γ T 3 nr T γ T c T T 4 T T 3 nrγ T > 0, 50 T T 3 the same esult as the clockwise cycle; thus it can also be expessed in the fom 46. Finally, as in the Otto cycle, condition T 4 <T is equivalent to W cycle >T h Sunivese Joule cycle, so we can make the same intepetation as the pevious sections. 3 These coefficients of pefomance can also be witten as Joule cycle γ /γ p γ /γ p γ /γ. p γ /γ p and COP R Joule cycle

20 Some consideations about themodynamic cycles 3 Pessue adiabatics T max 3 4 T min V c V V Volume Figue 9. Volume pessue diagam of the Diesel cycle. 8. Diesel cycle Let us now conside the Diesel cycle 4 shown in figue 9, with two adiabatic pocesses, one isobaic pocess and one isochoic pocess, whee > c >. The paamete c is called the cut-off atio 5. Note that, in contast to all pevious cycles, this cycle contains thee diffeent kinds of pocesses instead of two. The points of the cycle with the highest and lowest tempeatues ae also shown. Since p p 3, we can wite T T 3 T 3 V 3 c. 5 V V 3 T V Since adiabatics can be descibed by TV γ constant, we have γ T V γ T V γ T T V V γ T 3 T T 3 T T T c γ γ T 3 V γ 3 T 4 V γ 4 T 4 T 3 V3 V 4 c 5 γ T 4 T T 4 T 3 T 3 T c γ. Thus, T 3 T T T3 T T T T c γ γ T c γ T 4 T T T4 γ T T c. Let us assume that the cycle follows the clockwise diection. In pocesses and 3 4, the gas is themally isolated; in pocess 3, the gas is in contact with a themal esevoi at tempeatue T h T 3, and in pocess 4, the gas is in contact with a themal esevoi 4 Used by Rudolph Chistian Kal Diesel in 893, in his intenal combustion engine. 5 Hee the compession atio is V V and the expansion atio is V 4 V 3 c <. Fo this cycle we can define γ γ γ two pessue atios: p p p V V γ and p p 3 p 4 V4 V 3 c γ. 53

21 3 MFFdaSilva at tempeatue T c T.Wehave Q 0 ; Q 3 nc p T 3 n γr T 3 T nrt γ c γ ; Q 34 0 ; Q 4 nc V T 4 n R T 4 T nrt γ c ; Q h Q 3 nrt γ c γ > 0, Q c Q 4 nrt γ c < 0, Q cycle Q h + Q c nrt [ γ c γ c γ ], W cycle Q cycle nrt [ γ c γ c γ ]. It is not difficult to show 6 that Q cycle > 0, so W cycle < 0. Thus the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; these popeties chaacteize a heat engine. The efficiency of a Diesel engine is given by ɛ Diesel cycle W cycle Q h nrt γ [ γ c γ γ c ] c nrt γ γ, 54 c γ γ c γ and can be put in the fom 7 ɛ Diesel cycle [ γ ] c < γ γ c ɛ γ Otto cycle ; thus, the efficiency of a Diesel engine is less than the efficiency of an Otto engine o a Joule engine with the same compession atio. The eason is clea: in the Diesel engine the expansion atio is smalle than in Otto o Joule engines. The entopy change of the univese afte one cycle of the Diesel engine is SDiesel univese cycle, Q h T h nr Q nrt γ c γ c γ + T c T 3 [ γc γ c γ ] nrt γ nr γ c T [ γc γ c ]. c γ 55 This depends only on the cut-off atio and on the natue of the gas; it is easy to veify 8 that SDiesel univese cycle, > 0, so the clockwise Diesel cycle is also ievesible. In ode to tansfom this heat engine into a heat pump/efigeato, we poceed as in Otto and Joule cycles: we invet the cycle, following the counteclockwise diection, and we use two new esevois, at tempeatues T 4 and T. In pocess 4, the gas is in contact with a esevoi at tempeatue T 4 ; in pocesses 4 3 and, the gas is themally isolated; in pocess 3, the gas is in contact with a esevoi at tempeatue T. We obtain γ c ; Q 4 Q 4 nrt Q 43 0 ; Q 3 Q 3 nrt γ c γ ; Q 0. 6 Since > c, we can wite γ c γ c >γ c c γ c. So we must study the function f x γ x x γ x fo x>, with γ>. 7 The inequality γ c γ > is poved by studying the function f c x x γx fo x>, with γ>. 8 We must study the function f 3 x x γ x fo x>, with γ>.

22 Some consideations about themodynamic cycles 33 Imposing now the condition T c T 4 <T T h, we will have Q h Q 3 nrt γ c γ < 0, Q c Q 4 nrt γ c > 0, Q cycle Q h + Q c nrt [ γ c γ c γ ] < 0, W cycle Q cycle nrt [ γ c γ c γ ] > 0. It should be noted that condition T 4 <T is equivalent to c γ < γ ; once we fix the values of c and γ, this condition establishes a minimum value fo the compession atio. The coefficients of pefomance of this inveted Diesel cycle ae Diesel cycle Q h W cycle COP R Diesel cycle Q c W cycle nrt γ nrt γ nrt γ γ c γ [ γ c γ c ] >, 56 γ c γ c γ γ c γ nrt [ γ c γ c γ c ] γ c γ γ c γ c γ > These two coefficients veify the geneal elation 9 and Diesel cycle. 58 ɛ Diesel cycle The entopy change of the univese afte one counteclockwise Diesel cycle is SDiesel univese cycle, Q h T h Q nrt γ c γ c γ nrt γ γ c T c T T 4 nr [ γ c γ ] γ c γ c γ nr [ γ c ] c γ 59 and is not equal to the esult obtained fo the clockwise cycle. This non-equality could be elated to the fact, obseved at the beginning of the analysis of this cycle, that the two non-adiabatic pocesses ae of diffeent kinds one isobaic, the othe isochoic. Anyway, SDiesel univese cycle, depends only on the cut-off atio and on the natue of the gas, and it is easy to veify 9 that SDiesel univese cycle, > 0, so the counteclockwise Diesel cycle is also ievesible. Let us calculate the diffeence between expessions 55 and 59. We have [ SDiesel univese cycle, Sunivese Diesel cycle, c γ + γ γ c + γ + ] c nr nr [ γ c γ c ] γ c c and it can be poved 0 that this diffeence is always positive, so SDiesel univese cycle, > Sunivese Diesel cycle,, 6 that is, the clockwise cycle epesenting Diesel engine geneates moe entopy than the counteclockwise cycle epesenting Diesel heat pump/efigeato. 9 Analysing the function f 4 x γx x fo x>, with γ>. γ 0 Studying the function f 5 x x x γ γ x x fo x>, with γ>. γ c 60

23 34 MFFdaSilva Pessue adiabatics T max 3 4 T min V V V Volume Figue 0. Volume pessue diagam of the Atkinson cycle. Finally, compaing the expessions of W cycle and T h Sunivese Diesel cycle,, we ealize that condition W cycle >T h Sunivese Diesel cycle, is equivalent to γ c < γ, which, as we saw, is the same as T 4 <T, the necessay condition to tansfom a Diesel engine, by invesion, into a heat pump/efigeato. Thus, as in all pevious cycles, we can intepet the value of T h Sunivese Diesel cycle, as the minimum wok we have to do on the gas to obtain a Diesel heat pump/efigeato. 9. Atkinson cycle Let us conside the Atkinson cycle shown in figue 0, with two adiabatic pocesses, one isochoic pocess and one isobaic pocess, whee >>; clealy, is the compession atio and is the expansion atio. Depending on the souce, this cycle is also known as the Sagent cycle o Humphey cycle. Simila to the Diesel cycle, this cycle contains thee diffeent kinds of pocesses instead of two. The points of the cycle with the highest and lowest tempeatues ae also shown. Since p p 4, we can wite T V T 4 V 4 T 4 T V 4 V. 6 As in the Diesel cycle, we also have hee two pessue atios: p p p γ and p p 3 p 4 γ. The designation Atkinson cycle [5] has a histoical explanation, because the cycle was used by James Atkinson in his intenal combustion engine, in 88. The name Sagent cycle [6, 7] is in honou of Chales Elliotte Sagent 86?, who used it in vaious gas engines he patented fom 905 on. The designation Humphey cycle [7] is based on a pump, patented by Hebet Albet Humphey in 906, that implemented this cycle with a gaseous mixtue and whee the wok was used to pump wate; this designation is also vey common in the liteatue concening pulse detonation engines [8, 9].

24 Some consideations about themodynamic cycles 35 Since adiabatics can be descibed by TV γ constant, we have T V γ T V γ T γ V γ T V T 3 V γ 3 T 4 V γ 4 T γ 3 V4 γ T 3 T 3 T 4 γ T 4 V 3 T T 4 T, 63 so T3 T 3 T T T γ γ γ T γ T T T T4 T 4 T T T T. 64 T Let us assume that the cycle is pefomed in the clockwise diection. In pocesses and 3 4, the gas is themally isolated; in pocess 3, the gas is in contact with a themal esevoi at tempeatue T h T 3 ; and in pocess 4, the gas is in contact with a themal esevoi at tempeatue T c T.Wehave Q 0 ; Q 3 nc V T 3 n R T 3 T nrt γ γ ; Q 34 0 ; Q 4 nc p T 4 n γr T 4 T nrt γ ; Q h Q 3 nrt γ γ > 0, Q c Q 4 nrt γ < 0, Q cycle Q h + Q c nrt γ γ γ > 0, W cycle Q cycle nrt γ γ γ < 0. Since the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok, this epesents a heat engine. The efficiency of Atkinson engine is given by ɛ Atkinson cycle Q c Q h and can be witten in the fom 3 γ ɛ Atkinson cycle o in the fom 4 ɛ Atkinson cycle nrt γ γ nrt γ γ γ γ [ γ ] γ γ γ [ γ ] γ γ γ γ, 65 γ γ γ γ < γ ɛ Canot cycle > γ ɛ Otto cycle. 3 The inequality esults fom γ γ >, easily poved though the function f 4 x of a pevious footnote fo x>, with γ>. 4 The inequality esults fom γ x>, with γ>. γ <, which is poved though the function f x of a pevious footnote fo

25 36 MFFdaSilva Thus, an Atkinson engine is less efficient than a Canot engine functioning between two esevois at maximum and minimum tempeatues. Nevetheless, it is moe efficient than a Otto engine o a Joule engine with the same compession atio. Why? Because the expansion atio is geate in Atkinson engine than in Otto o Joule engines. Combining these esults with those obtained in the pevious sections, we obtain 0 <ɛ Diesel cycle <ɛ Otto cycle ɛ Joule cycle <ɛ Atkinson cycle <ɛ Canot cycle <. The entopy change of the univese afte one cycle of the Atkinson engine is nrt γ γ SAtkinson univese cycle, Q h Q nrt γ γ c γ + T h T c T 3 T nr [ ] γ γ, 66 γ an expession that depends only on the atio and on the natue of the gas; it can be shown that SAtkinson univese cycle, > 0, so the clockwise Atkinson cycle is also ievesible. In ode to tansfom this heat engine into a heat pump/efigeato, we poceed as in Otto, Joule and Diesel cycles: we invet the cycle to the counteclockwise diection, and we use two new esevois, at tempeatues T 4 and T. In pocess 4, the gas is in contact with a esevoi at tempeatue T 4 ; in pocesses 4 3 and, the gas is themally isolated; in pocess 3, the gas is in contact with a esevoi at tempeatue T. We obtain Q 4 Q 4 nrt γ ; Q 43 0 ; Q 3 Q 3 nrt γ γ ; Q 0. Imposing the condition T c T 4 <T T, we will obtain the following fomulae: h Q h Q 3 nrt γ γ < 0, Q c Q 4 nrt γ Q cycle Q h + Q c nrt γ γ γ < 0, W cycle Q cycle nrt γ γ γ > 0. > 0, Simila to what occued in the pevious section, condition T 4 <T is equivalent to γ > ; once we fix the values of and γ, this condition establishes a minimum value fo the compession atio. The coefficients of pefomance of this inveted Atkinson cycle ae Atkinson cycle Q h W cycle COP R Atkinson cycle Q c W cycle nrt γ nrt γ nrt γ γ γ γ γ γ nrt γ γ γ γ γ >, 67 γ γ γ γ γ γ γ γ γ > 0, 68 veifying the geneal elation 9 and Atkinson cycle. 69 ɛ Atkinson cycle Taking into account the pevious compaison between the efficiencies of the vaious cycles, and combining elations 7, 40, 49, 58 and 69, we would obtain the following esult: < Canot cycle < COPHP Atkinson cycle < COPHP Otto cycle COPHP Joule cycle < COPHP Diesel cycle.

26 Some consideations about themodynamic cycles 37 The second inequality seems to violate the Canot theoem: appaently, we would have heat pumps with a coefficient of pefomance geate than the coefficient of pefomance of a Canot heat pump. But thee is no violation at all: we should note that Atkinson, Otto, Joule and Diesel heat pumps, appeaing in the pevious expession, function between esevois at tempeatues T c T 4 <T T h, wheeas the Canot heat pump, with which they ae being compaed, functions between esevois at tempeatues T c T <T 3 T h. If we compae Diesel cycle with the coefficient of pefomance of a Canot heat pump functioning between esevois at tempeatues T 4 and T, we veify that the latte is geate, accoding to Canot theoem, and we obtain we leave the details to the eade < Atkinson cycle < COPHP Otto cycle COPHP Joule cycle < COPHP Diesel cycle < COPHP Canot cycle. The entopy change of the univese afte one counteclockwise Atkinson cycle is S univese Atkinson cycle, Q h T h nr Q c T c nrt γ γ γ T [ γ γ γ γ nrt γ γ ] T 4 nr [ γ γ ] 70 and is not equal to the esult obtained fo the clockwise cycle. Once again, this non-equality should be elated to the fact that the two non-adiabatic pocesses ae of diffeent kinds one isobaic, the othe isochoic. Anyway, SAtkinson univese cycle, depends only on the atio and on the natue of the gas, and it is easy to veify 5 that SAtkinson univese cycle, > 0, which shows that the counteclockwise Atkinson cycle is also ievesible. Let us calculate the diffeence between expessions 66 and 70. We have SAtkinson univese cycle, Sunivese Atkinson cycle, nr [ γ ] + γ γ γ ++γ γ γ nr [ γ γ γ γ γ ] 7 and it can be shown 6 that this diffeence is always negative, so SAtkinson univese cycle, < Sunivese Atkinson cycle,, 7 that is, the clockwise cycle epesenting Atkinson engine geneates less entopy than the counteclockwise cycle epesenting Atkinson heat pump/efigeato. To conclude this section, if we compae expessions of W cycle and T h Sunivese Atkinson cycle,,we ealize that condition W cycle >T h Sunivese Atkinson cycle, is equivalent to γ >, which, as we saw, is the same as T 4 <T, the necessay condition to tansfom an Atkinson engine, by invesion, into a heat pump/efigeato. Thus, as in all pevious cases, we can intepet the quantity T h Sunivese Atkinson cycle, as the minimum wok we have to do on the gas to implement an Atkinson heat pump/efigeato. In this and all pevious cycles, we saw that the invesion of a heat engine cycle, made unde cetain conditions, geneated the cycle of a heat pump/efigeato; conditions which allow that tansfomation wee always expessed in tems of one o moe paametes, and it was always possible to intepet them as the minimum wok we have to bing to the system. Thus it would seem easonable to think that the tansfomation of a heat engine cycle into a heat pump/efigeato cycle is always possible unde specified conditions. But that idea is wong, as we will show in the following section. 5 Analysing the function f 3 x of a pevious footnote fo x>, with γ>. 6 Analysing the function f 5 x of a pevious footnote fo x>, with γ>.

27 38 MFFdaSilva Pessue p p T max 3 p 4 T min V V Volume Figue. Volume pessue diagam of the ectangula cycle. 0. Rectangula cycle Let us now conside the ectangula cycle shown in figue, with two isochoic pocesses and two isobaic pocesses, whee, p > could be named compession o expansion atio, and p could be named pessue atio. Minimum and maximum tempeatues occu at points and 3, espectively. Although this cycle shows similaities with some of the pevious cycles, we should note that it does not include any isothemal o adiabatic pocess. Let us assume that the cycle follows the clockwise diection. In pocesses 3, the gas is in contact with a themal esevoi at tempeatue T h T 3, and in pocesses 3 4, the gas is in contact with a themal esevoi at tempeatue T c T.Wehave Q nc V T n R T T ppv pv Q 3 nc p T 3 n γr T 3 T γ ppv p pv Q 34 nc V T 34 n R T 4 T 3 pv ppv Q 4 nc p T 4 n γr T T 4 So γpv pv p pv ; γ p pv ; p pv ; γ pv. Q h Q + Q 3 p + γ p pv > 0, Q c Q 34 + Q 4 p + γ pv < 0, Q cycle Q h +Q c p pv > 0, W cycle Q cycle p pv < 0.

28 Some consideations about themodynamic cycles 39 Thus, the gas is absobing heat fom the esevoi at highe tempeatue, is ejecting heat to the esevoi at lowe tempeatue and is doing wok; this clealy epesents a heat engine. The efficiency of this heat engine is given by ɛ ectangula cycle W cycle Q h p pv p +γ p pv p p + γ p γ and if we compae it with the efficiency of a Canot engine functioning between the same esevois, ɛ Canot cycle T nrt pv T 3 nrt 3 p pv p, 74 p p we find that ɛ ectangula cycle <ɛ Canot cycle because ɛ Canot cycle ɛ ectangula cycle p p + γ p p [ p + γ p ] > 0. The entopy change of the univese afte one cycle of this heat engine is Sectangula univese cycle, Q h Q c T h nr p +γ p γ p pv nr pv p +γ γ pv nr pv + T c [ p + γ p + γ p p nr p p + γ p > 0, 75 p showing the ievesible chaacte of this cycle. Let us ty, as in the examples of pevious sections, to tansfom this heat engine into a heat pump/efigeato. In ode to do so, we invet the ectangula cycle, pefoming it in the counteclockwise diection. In pocesses 4 3, the gas is in contact with a themal esevoi at tempeatue T h T h T 3, and in pocesses 3, the gas is in contact with a themal esevoi at tempeatue T c T c T.Wehave γ Q 4 Q 4 pv ; Q 43 Q 34 p pv ; Q 3 Q 3 γ p pv ; Q Q p pv, so Q h Q 4 + Q 43 p + γ pv > 0, Q c Q 3 + Q p + γ p pv < 0, Q cycle Q h + Q c p pv < 0, W cycle Q cycle p pv > 0. Thus, the gas is still absobing heat fom the esevoi at highe tempeatue, it is still ejecting heat to the esevoi at lowe tempeatue, but now wok is being done on the gas; this situation does not coespond to a heat engine, no a heat pump/efigeato. Its schematic epesentation is shown in figue 4a. So it makes no sense hee to calculate the efficiency o the coefficients of pefomance. ] 73