A critique of some modern applications of the Carnot heat engine concept: the dissipative heat engine cannot exist

Transcription

1 doi: /rsp Published online A critique of some modern pplictions of the Crnot het engine concept: the dissiptive het engine cnnot exist BY ANASTASSIA M. MAKARIEVA 1, *, VICTOR G. GORSHKOV 1,BAI-LIAN LI 2 AND ANTONIO DONATO NOBRE 3 1 Theoreticl Physics Division, Petersburg Nucler Physics Institute, Gtchin, St Petersburg , Russi 2 Ecologicl Complexity nd Modeling Lbortory, Deprtment of Botny nd Plnt Sciences, University of Cliforni, Riverside, CA , USA 3 Instituto Ncionl de Pesquiss d Amzôni, Mnus AM , Brzil In severl recent studies, het engine operting on the bsis of the Crnot cycle is considered, where the mechnicl work performed by the engine is dissipted within the engine t the temperture of the wrmer isotherm nd the resulting het is dded to the engine together with n externl het input. This internl dissiption is supposed to increse the totl het input to the engine nd elevte the mount of mechnicl work produced by the engine per cycle. Here it is rgued tht such dissiptive het engine violtes the lws of thermodynmics. The existing physicl models employing the dissiptive het engine concept, in prticulr the het engine model of hurricne development, need to be revised. Keywords: dissiptive het engine; Crnot cycle; dissiption; efficiency 1. Introduction The Crnot cycle does not involve irreversible processes or dissiptive losses. During one cycle in Crnot het engine, the working body ( fluid cpble of expnsion, usully gs) receives het Q h from hot body (the heter) nd performs mechnicl work A h = Q h t temperture T h. It gives het Q c wy to cold body (the cooler) while work A c = Q c Q h is performed on the gs t temperture T c T h. The resulting work A > 0 is determined by the energy conservtion lw (the first lw of thermodynmics) s A = Q h Q c. This work is performed by the working body on its environment. Since ll the processes in the Crnot cycle re reversible, entropy of the working body is conserved. Entropy of the environment on which the work is performed does not chnge either. The mount of entropy S h = Q h /T h received from the heter is equl to the mount of entropy S c = Q c /T c given wy to the cooler. The equlity Q h /T h = Q c /T c, which stems from the second lw of thermodynmics, *Author for correspondence (elb@peterlink.ru). Received 6 November 2009 Accepted 10 December This journl is 2010 The Royl Society

2 2 A. M. Mkriev et l. combines with the energy conservtion lw to determine efficiency 3 A/Q h of the Crnot cycle s 3 = (T h T c )/T h < 1. For the Crnot het engine, A Q h. The dissiptive het engine concept dvnced by Rennó & Ingersoll (1996) nd discussed by Rennó (1997, 2001), Puluis et l. (2000) nd Puluis & Held (2002) is used to ccount for hurricne intensity (Bister & Emnuel 1998; Emnuel 2003). In the dissiptive het engine, work A d (or some prt of it in the generl cse) produced in the idel Crnot cycle undergoes dissiption t temperture T h of the heter. (Subscript index d stnds for the dissiptive het engine.) The resulting het is dded to the working body together with the externl het Q hd tht comes from the heter. In the sttionry cse, the reltionship between work A d nd externl het Q hd is then written s A d = 3(Q hd + A d ). Efficiency 3 d = A d /Q hd of the dissiptive het engine becomes 3 d = (T h T c )/T c. Thus, for given Q hd, the efficiency of the dissiptive het engine grows infinitely with decresing T c, nd work A d cn become much lrger thn Q hd : A d Q hd t T c T h T c nd 3 d 1; A d t T c 0. Demnding energy to be conserved gives Q hd = Q c, i.e. the mount of het received by the dissiptive het engine from the heter coincides with the mount of het disposed to the cooler. It is ssumed tht when work A d dissiptes within the working body of the dissiptive het engine in contct with the heter, i.e. t T = T h, entropy increses by S hd = (Q hd + A d )/T h. The decrese in entropy due to contct with the cooler remins S c = Q c /T c s in the Crnot het engine. Tking into ccount tht Q h = Q c nd A d = 3 d Q hd, the mthemticl equlity S hd = S c holds. From this, it is concluded tht in the dissiptive het engine the entropy of the working body remins constnt nd tht the dissiptive het engine conforms to both first nd second lws of thermodynmics. The min feture of the dissiptive het engine is the increse in the work produced by the engine per cycle due to internl dissiption compred with the sme engine without dissiption. It is stted tht the frction of mechnicl energy dissipted... increses the het input to the convective het engine (Rennó & Ingersoll 1996, p. 579), so tht more energy is vilble to be converted into mechnicl energy (Rennó & Ingersoll 1996, p. 578). In other words, by dissipting work within the engine, it is supposed to be possible to increse the per-cycle output of the mechnicl work. In this pper, we show tht this concept of the dissiptive het engine is bsed on physicl misinterprettion of the nture of the Crnot het engine. When the essentil physicl fetures of the het engine re tken into ccount, the concept of the dissiptive het engine is shown to be in conflict with the lws of thermodynmics. 2. Physics of the Crnot het engine The Crnot cycle consists of two isotherms t tempertures T = T h nd T = T c of the heter nd the cooler, respectively, nd of two dibtes connecting the isotherms. The working body in thermodynmic equilibrium with the heter t T = T h cnnot receive het from the ltter. To receive het, the working body must expnd first, so tht its temperture becomes little lower thn tht of the heter, only then the het flux from the heter to the working body becomes

3 Dissiptive het engine: critique Figure 1. A Crnot het engine. The working body (gs) (open circles) is contined within closed cylinder (1) with sliding piston (2) s the cp. Spring (3) mkes the piston move. Elstic stops (4) limit the minimum nd mximum volumes of the gs in the cycle. During the cycle, the working body is sequentilly brought into contct with the heter (5) t T = T h nd the cooler (6) t T = T c. The environment where the engine works (7) is ssumed to be infinite, so its pressure p is constnt during the cycle. possible. Thus, the Crnot het engine operting on the bsis of the Crnot cycle must be furnished with n uxilliry dynmic device tht performs mechnicl expnsion nd contrction of the working body. A Crnot het engine where the role of such device is plyed by n idel elstic spring is shown in figure 1. The working body (gs) is contined in the cylinder cpped on one side by sliding piston tht is connected to the spring. The piston trvels within the cylinder without friction. Two stops re provided to limit the piston s movement nd to define the minimum nd mximum volumes occupied by the working body (figure 1). In the ultimte sttes of mximum compression nd extension of the spring its potentil energy is mximum. In the intermedite stte where the spring is relxed, its potentil energy is zero, while the kinetic energy of the working body nd the piston is not. The Crnot het engine is put into opertion by introducing n mount of potentil energy into the engine. We will cll this energy the strt-up energy. For exmple, when the gs hs the minimum volume in contct with the heter, the spring is extended to mximum (figure 2). In this cse, the strt-up energy hs the form of the potentil energy of the extended spring. The strt-up energy cn tke other forms. For exmple, it cn be dded s surplus pressure of the working body compred with the environment or s the kinetic energy of the working body nd the piston. The cycle strts when the spring is extended to its utmost. At this moment the gs occupies the smllest volume (point in figure 2) nd gs pressure in the cylinder is equl to tht of the externl environment. The cylinder, which idelly hs n infinite het conductivity, is put in contct with the heter t T = T h. The spring strts compressing nd moves the piston to the right such tht the volume occupied by the working body increses t constnt temperture (the wrmer isotherm of the Crnot cycle). The working body receives het nd, together with the spring, performs mechnicl work on moving the piston. After the mount of received het reches Q h, point b in figure 2b, the contct with the heter is mechniclly disrupted. The gs further expnds dibticlly until its temperture diminishes from T h to T c, point c in figure 2c. At this point, the working body is brought into contct with the cooler t T = T c.

4 4 A. M. Mkriev et l. (b) b () (c) c (d ) d Figure 2. Crnot cycle. () Beginning of the wrmer isotherm t T = T h ; the working body is brought into contct with the heter, the spring pulls the piston to the right nd the gs expnds. The initil pressure p of gs t point coincides with tht of the externl environment. (b) End of the wrmer isotherm, beginning of the first dibte; the heter is detched from the engine, the gs expnds dibticlly nd its temperture drops from T = T h to T = T c.(c) End of the first dibte, beginning of the colder isotherm t T = T c ; the working body is brought into contct with the cooler, the spring extends nd pushes the piston to the left. (d) End of the colder isotherm, beginning of the second dibte; gs is compressed by the moving piston nd the gs temperture increses from T = T c to T = T h. Note tht t the end of the cycle the piston hs cquired kinetic energy equl to net work A performed by the working body in the cycle. To keep the engine sttionry, this energy should be tken wy from the engine t point. Note tht, in order for the ssocited ir velocity to be sufficiently low so s not to disturb the thermodynmic equilibrium, the piston should be mde sufficiently hevy. The piston velocity t point c becomes zero; the spring strts extending nd compresses the working body, which llows the ltter to dispose het to the cooler. At point d in figure 2d, the mount of disposed het reches Q c, the cooler is detched from the cylinder nd the gs continues to be compressed dibticlly. Its temperture rises bck to T h, point in figure 2. At this point, the Crnot cycle is completed. Work A performed by the gs hs tken the form of the kinetic energy of the piston nd cn be used outside the engine. If this work is not tken wy, it will continuously ccumulte within the engine, incresing the piston velocity nd kinetic energy with ech cycle. As result, the power of the engine (i.e. the number of cycles per unit time) will increse. Two spects need to be emphsized. First, the strt-up energy is principlly importnt for the het engine to operte. If there is no spring, nd the gs in the cylinder is in thermodynmic equilibrium with the heter nd with the environment, the piston will remin immobile, the engine will not operte nd no work will be produced. In the cse of n infinite environment (figure 1),

5 Dissiptive het engine: critique 5 in which pressure p does not chnge during the cycle, the necessry mount of the strt-up energy E (J mol 1 ) cn be clculted s the difference between work A p performed by the piston on the environment with constnt pressure p nd work A w performed by the working gs with p(v) p on the piston s the piston moves from point to point c, E = A p A w = c (p p(v))dv = pdv c p(v)dv 0, where v is molr volume, pv = RT is the eqution of stte for the idel gs nd R is the universl gs constnt. It is esy to see tht, t Dv/v 1, Dv v b v, the strt-up energy E should be of the order of Q h = b p(v)dv = RT h ln(1 + Dv/v ). Indeed, we hve E pdv RT h ln(1 + Dv/v ) pdv RT h (Dv/v ) + RT h (Dv/v ) 2 /2 RT h /2tDv/v 1 nd p = p. This is conservtive estimte tht ignores work performed on the dibte b c where the working body continues to expnd. Second, if the het conductivity of the heter nd the cooler is sufficiently lrge, it strictly ensures constnt temperture s the piston moves from point to point b t T = T h nd from point c to point d t T = T c. Therefore, the mounts of het Q h nd Q c received nd given wy, respectively, by the working body re unmbigously determined by the construction of the engine. At given T h, the vlue of Q h is determined by the chnge of molr volume v from point to point b, Q h = b p dv pdv for smll reltive chnges of gs pressure p(v). The first nd second lws of thermodynmics for the Crnot cycle tke the form nd Q h A = Q c (2.1) Q h T h = Q c T c. (2.2) The five mgnitudes entering equtions (2.1) nd (2.2) leve three out of the five vribles independent, e.g. Q h, T h nd T c. Equtions (2.1) nd (2.2) cn be written s A = 3Q h, Q c = (1 3)Q h nd 3 T h T c. (2.3) T h Here two vribles re independent, Q h nd 3 tht depend on T h nd T c. From eqution (2.3), we obtin by replcing the independent vrible Q h by Q c A = Q 3 c or A = 3(Q c + A), 1 3 = T h T c. (2.4) T c It is noteworthy tht these reltionships coincide with those for the dissiptive het engine if one replces Q c by formlly introduced vrible Q hd Q c, (2.5) where Q hd refers to the externl het input in the dissiptive het engine. Note tht t T c 0 in eqution (2.5) work A remins finite in view of eqution (2.2): t constnt Q h nd T h, the decrese in T c must be ccompnied by proportionl decrese in Q c.

6 6 A. M. Mkriev et l. 3. Physicl mening of the mthemticl reltionships of the dissiptive het engine Thus, the mthemtics of the dissiptive het engine formlly rewrite the first nd second lws of thermodynmics for the Crnot het engine, equtions (2.1) nd (2.2), by introducing new vrible Q hd Q c. As shown in the previous section, work A cn be produced by the Crnot het engine fter it is supplied with the strt-up energy nd the working body receives het Q h from the heter. After work is performed in the first cycle, it cn, in principle, be dissipted into het t T = T h nd introduced into the working body t the first stge of the second cycle. We note tht, lthough principlly possible, such procedure is techniclly difficult, s it demnds resonnce between the dynmics of the piston movement nd tht of the dissiption process. Chrcteristic times of the piston movement nd the dissiption process re dictted by different physicl lws nd generlly do not coincide. Resonnce synchroniztion of independent physicl processes is in the generl cse impossible. (For exmple, it would need to be proved tht ll kinetic energy originting within hurricne dissiptes within the hurricne nd is not trnsported wy in the upper tmosphere to dissipte to het outside the hurricne re, i.e. tht this internl dissiption is synchronized with the supposed het input from the ocen (the heter). Without such proof, the ppliction of the min eqution of the dissiptive het engine, A d = 3(Q hd + A d )(Emnuel 2003), is not vlid.) When work A dissiptes to het within the working body, the ltter wrms. The Crnot het engine is originlly constructed such tht, while the piston moves from point to point b (figure 2), the working body receives het Q h. This is possible due to the fct tht, when the piston moves nd the volume of the gs increses, the gs becomes little colder thn the heter, enbling the necessry het to flow from the heter to the engine. If the working body now hs source of het inside, the extension of the working gs due to piston movement does not sufficiently decrese the gs temperture to ensure the sme flux Q h from the heter. As prescribed by the first lw of thermodynmics nd the idel gs eqution, gs tht isothermlly expnds by preset mount (from point to point b in figure 2) receives fixed mount of het Q h = b p dv. If some prt of this het Q A = A is delivered to the working body s the product of dissiption of work A, then the mount of externl het Q hd received by the engine from the heter will decrese compred with the Crnot het engine to Q hd = Q h A. Since the mount of het received by the working body remins unchnged, s dictted by the geometry of the working cylinder (figure 1), the mount of work A performed by the engine is lso constnt nd cnnot be incresed by dissipting work performed by the engine in the previous cycles. 4. The impossibility of the dissiptive het engine Thus, it is the centrl ide tht the frction of mechnicl energy dissipted... increses the het input to the convective het engine (Rennó & Ingersoll 1996, p. 579) is the min physicl inconsistency in the concept of the dissiptive het engine, which brings the concept into conflict with the lws of thermodynmics.

7 Dissiptive het engine: critique 7 Let Q hd = Q h be the totl mount of het received by the dissiptive het engine in the first cycle (no work previously produced). Work A 1 = A produced during this cycle is determined by Crnot eqution (2.3). In the subsequent cycles, het formed due to the dissiption of work produced in the preceding cycle is dded to the fixed mount of externl het Q hd received from the heter A n+1 = 3(Q hd + A n ) nd A 1 = A = 3Q hd, (4.1) where n 1 is the number of the current cycle. We hve from eqution (4.1) nd A n = 3(Q h ) n, (Q h ) n Q hd ( n 1 ) (4.2) (Q cd ) n = (Q h ) n A n = Q hd (1 3 n ). (4.3) Here, A n is work produced in the nth cycle; (Q h ) n is the totl het received by the working body (externl het plus the dissiptive het from work performed in the previous cycles) note tht, ccording to the dissiptive het engine concept, (Q h ) n grows with n following the increse in internlly dissipted work; (Q cd ) n is the mount of het given wy to the cooler, it decreses with growing n t 3 < 1. As is esy to see, t n work, A d = A is formlly determined by the eqution of the dissiptive het engine nd A d = 3(Q hd + A d ) nd Q cd = (1 3)(Q hd + A d ) = Q hd (4.4) A d = Q hd = T h T c Q hd. (4.5) T c A similr derivtion cn be performed for generl cse when not ll but some prt g 1ofworkA n produced in the cycle is dissipted t the wrmer isotherm, A n+1 = 3(Q hd + ga n ), cf. eqution (4.1), while the rest of the produced work, (1 g)a n, is dissipted t the colder isotherm with the resulting het disptched to the cooler (Rennó & Ingersoll 1996). We then obtin A d = 3 1 g3 Q T h T c hd = Q hd. (4.6) (1 g)t h + gt c In the cse of zero dissiption t the wrmer isotherm, g = 0 nd Q hd = Q h, eqution (4.6) coincides with eqution (2.3) for work A of the Crnot cycle. Equtions (4.5) nd (4.6) sy tht work A d performed by the dissiptive het engine increses infinitely t fixed Q hd with T c 0. Thus, the dissiptive het engine represents n engine tht recircultes het to work nd bck t potentilly infinite power; it produces work A d greter thn it would in the bsence of dissiption, i.e. greter thn the work A of the Crnot het engine tht opertes with the sme externl het input Q hd = Q h, cf. equtions (4.5) nd (2.3). An infinite power of this recircultion cn be chieved by simply decresing temperture T c of the cooler. Such n engine violtes both the first nd the second lws of thermodynmics nd cnnot exist. Indeed, since for given het engine Q h = Q hd + A d = const., work A n produced in ech cycle is lso constnt, A n = 3Q h = A. This mens tht no ccumultion of mechnicl work beyond A within the engine is possible in ny cycle. Thus,

8 8 A. M. Mkriev et l. the property A d > A of the dissiptive het engine would violte the energy conservtion lw (the first lw of thermodynmics), becuse the difference A d A > 0 remins unccounted for by the energy blnce. In order to increse the het input to the engine, it is necessry to reconstruct it. At constnt T h nd T c, such reconstruction would imply, first, n increse in the strt-up energy nd, second, n increse in the liner size of the engine to llow for greter expnsion of the working body t the wrmer isotherm (e.g. Leff 1987). Internl dissiption of produced work within the engine does not led to n increse in the het input. In the Crnot cycle, ll het Q h received by the working body on the wrmer isotherm is converted to work with n entropy increse S h = Q h /T h = ( b p dv)/t h tht corresponds to the isotherml gs expnsion s dictted by the construction of the engine. Thus, the sme condition Q hd + A d > Q h of the dissiptive het engine would men n dditionl dissiption of work A d to het nd its regenertion bck to work A d from het t one nd the sme temperture, which is prohibited by the second lw of thermodynmics ds dq/t. Indeed, for the wrmer isotherm, we would then hve insted b DS = p dv T h DQ T h = Q hd + A d T h. (4.7) 5. Conclusions nd discussion () Internl dissiption cnnot increse the work output of het engine Equtions (4.5) nd (4.6) for work A d of het engine where the work produced is dissipted within the engine re incorrect. The problem consists in the fct tht with incresing dissiption rte A n the externl het input in equtions (4.1) nd (4.4) does not remin constnt, but decreses s Q hd = Q h A n, while totl het input Q h to the working body remins constnt s prescribed by the construction of the engine in question. Putting Q hd = Q h A d into eqution (4.5) gives Q hd = (1 3)Q h = (T c /T h )Q h. Therefore, t T c 0, we hve Q hd 0 (no het input from the heter) nd work A d = A = 3Q h = const. remins limited nd equl to tht of Crnot het engine where no dissiption tkes plce. We summrize tht dissiption of work within het engine cnnot increse the work produced by the engine. In the generl cse, work A = A d = const. produced by the engine does not depend on the proportion g of mechnicl work dissipted t the wrmer isotherm. Eqution (4.6) is physiclly misleding, s it hides the dependence of Q hd on g nd Q h, Q hd = Q h ga d, which, when entered into the eqution, yields A d = 3Q h = const.(g). It is importnt to note tht, s fr s Q h = Q hd + A d = const., the entropy blnce eqution for the dissiptive het engine S d = Q hd /T h + A d /T h Q c /T c = Q h /T h Q c /T c = 0 is mthemticlly identicl to the entropy blnce eqution of the Crnot het engine, where one term Q h /T h is formlly divided into two, Q h /T h Q hd /T h + A d /T h. The problem with the dissiptive het engine is physicl, not mthemticl. It could not hve been reveled from forml considertion of the engine s entropy budget without considering the essentil physicl peculirities of wht het engine is.

9 Dissiptive het engine: critique 9 (b) Relted problems in pplying the het engine concept to the tmosphere In the mentime, the vilble theoreticl considertions of tmospheric circultion on the bsis of thermodynmic cycle such s the Crnot cycle (Rennó & Ingersoll 1996; Rennó 1997, 2001; Puluis et l. 2000; Puluis & Held 2002; Lucrini 2009) concentrte on evluting the energy nd entropy budgets but refrin from ddressing the dynmic nd structurl spects of the het engine s physicl entity chrcterized by inherent sptil nd temporl scles. Nmely, this hs cused the dissiptive het engine inconsistency discussed in this pper, but the conceptul problem ppers to be wider nd wrrnts further investigtions. In prticulr, the necessity for n uxilliry dynmic system tht would possess the strt-up energy E to expnd nd contrct the working body of the tmospheric het engine, the ir, is not tken into ccount. The clssicl Crnot het engine is bsed on equilibrium thermodynmics, which prescribes tht ll the non-equilibrium processes of the cycle such s het trnsfer from the heter to the working body occur t n infinitely smll rte. Therefore, the power (work performed per unit time) of the idel Crnot het engine is infinitely smll. Efficiency 3 mp corresponding to mximum power output of rel het engines is not equl to Crnot efficiency, but pproches 3 mp = 1 T c /T h s first pointed out by Curzon & Ahlborn (1975). This result ws confirmed by rigorous theoreticl investigtions recently performed in vrious domins of science (Leff 1987; Rebhn 2002; Vn den Broeck 2005; Feidt et l. 2007; Jiménez de Cisneros & Clvo Hernández 2008; Izumid & Okud 2009). This result ws, however, neglected in the formultion of the mximum potentil intensity model for hurricnes bsed on the het engine concept (e.g. Emnuel 2003). This points to brech between the tretments of the het engine concept in modern climtology versus theoreticl physics. A working body in thermodynmic equilibrium with n infinite het source (ocen) cnnot spontneously strt expnding nd receiving het t finite rte; this is thermodynmiclly prohibited. As is known from the studies of finitetime thermodynmics of het engines tht hve been recently gining momentum (Rebhn 2002; Vn den Broeck 2005; Feidt et l. 2007; Izumid & Okud 2009), the time scle for opertion of het engines producing finite power is set externlly by the uxilliry system (e.g. the mechnism tht moves the piston t given velocity). As the piston is externlly moved, the temperture of the gs is lowered nd there ppers to be temperture difference enbling het flow from the hot reservoir to the working body. This llows for finite rte of het flow to the engine. This temperture difference grows with incresing piston velocity (Izumid & Okud 2009), while the engine efficiency decreses. It is not tht some prts of the engine move becuse the engine receives het from the heter, but, conversely, the engine cn receive het only becuse some prts of the engine re moved by the uxilliry mechnicl system. In the tmosphere, no independent physicl mechnism hs ever been identified tht would compress nd decompress the ir nd trnsport it between the cold nd hot reservoirs (e.g. between the surfce nd upper troposphere, s in the hurricne model of Emnuel (2003)) t finite velocity in mnner similr to how it is done by the piston-moving mechnism in rel het engines. Unless such mechnism is described, the horizontl drop in ir pressure observed in hurricnes cnnot be explined s the outcome of Crnot het engine operting in the tmosphere. The need for n independent specifiction of horizontl pressure

10 10 A. M. Mkriev et l. grdient ws identified s fundmentl problem of the thermodynmic pproch to hurricne formtion (Smith et l. 2008). In prllel, it ws recently proposed tht the nture of tmospheric circultion is dynmic, not thermodynmic, nd reltes to the relese of potentil energy during condenstion of wter vpor (Mkriev & Gorshkov 2007, 2009,b). In summry, the problem of pplicbility of the het engine concept to the tmosphere, to which the present nlysis hs imed to contribute, ppers to justify the broder ttention of scientists from different fields. We thnk Dr Eckhrd Rebhn for constructive criticisms of n erlier version of the pper. Helpful comments of two nonymous referees re grtefully cknowledged. References Bister, M. & Emnuel, K. A Dissiptive heting nd hurricne intensity. Meteorol. Atmos. Phys. 65, (doi: /bf ) Curzon, F. L. & Ahlborn, B Efficiency of Crnot engine t mximum power output. Am. J. Phys. 43, (doi: / ) Emnuel, K. J Tropicl cyclones. Annu. Rev. Erth Plnet. Sci. 31, (doi: /nnurev.erth ) Feidt, M., Coste, M., Petre, C. & Petrescu, S Optimiztion of the direct Crnot cycle. Appl. Therm. Eng. 27, (doi: /j.pplthermleng ) Izumid, Y. & Okud, K Onsger coefficients of finite-time Crnot cycle. Phys. Rev. E 80, (doi: /physreve ) Jiménez de Cisneros, B. & Clvo Hernández, A Coupled het devices in liner irreversible thermodynmics. Phys. Rev. E 77, (doi: /physreve ) Leff, H. S Therml efficiency t mximum work output: new results for old het engines. Am. J. Phys. 55, (doi: / ) Lucrini, V Thermodynmic efficiency nd entropy production in the climte system. Phys. Rev. E 80, (doi: /physreve ) Mkriev, A. M. & Gorshkov, V. G Biotic pump of tmospheric moisture s driver of the hydrologicl cycle on lnd. Hydrol. Erth Syst. Sci. 11, Mkriev, A. M. & Gorshkov, V. G Condenstion-induced dynmic gs fluxes in mixture of condensble nd non-condensble gses. Phys. Lett. A 373, (doi: / j.physlet ) Mkriev, A. M. & Gorshkov, V. G. 2009b Condenstion-induced kinemtics nd dynmics of cyclones, hurricnes nd torndoes. Phys. Lett. A 373, (doi: /j.physlet ) Puluis, O. & Held, I. M Entropy budget of n tmosphere in rditive-convective equilibrium. Prt I. Mximum work nd frictionl dissiption. J. Atmos. Sci. 59, (doi: / (2002)059%3c0125:eboaai%3e2.0.co;2) Puluis, O., Blji, V. & Held, I. M Frictionl dissiption in precipitting tmosphere. J. Atmos. Sci. 57, (doi: / (2000)057%3c0989:fdiapa%3e2.0.co;2) Rebhn, E Efficiency of nonidel Crnot engines with friction nd het losses. Am. J. Phys. 70, (doi: / ) Rennó, N. O Reply: remrks on nturl convection s het engine. J. Atmos. Sci. 54, (doi: / (1997)054%3c2780:rronca%3e2.0.co;2) Rennó, N. O Comments on Frictionl dissiption in precipitting tmosphere. J. Atmos. Sci. 58, (doi: / (2001)058%3c1173:cofdia%3e2.0.co;2) Rennó, N. O. & Ingersoll, A. P Nturl convection s het engine: theory for CAPE. J. Atmos. Sci. 53, (doi: / (1996)053%3c0572:ncaahe%3e2.0.co;2) Smith, R. K., Montgomery, M. T. & Vogl, S A critique of Emnuel s hurricne model nd potentil intensity theory. Q. J. R. Meteorol. Soc. 134, (doi: /qj.241) Vn den Broeck, C Thermodynmic efficiency t mximum power. Phys. Rev. Lett. 95, (doi: /physrevlett )