Maximum work for arnot-like eat engines wit infinite eat soure Rui Long and Wei Liu* Sool of Energy and Power Engineering, Huazong University of Siene and enology, Wuan 4374, ina orresponding autor: Wei Liu (w_liu@ust.edu.n) An analysis of effiieny and its bounds at maximum work output for arnot-like eat engines is onduted and te eat transfer proesses are desribed by Newton s law of ooling wit timedependent eat ondutane. e upper bound of te effiieny is found to be te A effiieny, and is independent of te time duration ompleting ea proess and te time-dependent ondutane. We prove tat even te working medium exanges eat suffiiently wit te eat 2 reservoirs, te work wi ould be extrated is finite and limited by W max+ m η. e imal temperature profiles in te eat exanging proesses are also analyzed. Wen te dimensionless ontat times satisfy ertain relations, te endoreversible model is reovered. PAS numbers: 5.7.-a, 5.7.Ln As we all know, te arnot effiieny, η 1 /, defines te upper bound of effiieny of all te eat engines operating between two eat reservoirs at temperatures and, > [1]. Any real-life eat engines sould operate at te effiieny lower tan η. However, in te arnot eat engines all te proesses are qusi-stati and te time for ompleting tat yle is infinitely long wi leads to zero power output. As to atual eat engines, its time duaraion is finite, wi sows a signifiant deviative from te ideal arnot ones. e ideal arnot yle must be speeded up to meet te atual demand. aking only into onsideration te irreversibility aused by eat transfer between te eat reservoirs and working substane during te isotermal proesses, urzon-alborn[2] proposed te onept of endoreversible arnot eat engine, and dedued its effiieny at maximum power output. at is te groundbreaking A effiienyη 1 A /. It opens te era of finite time termodynamis. In addition, Otto and J-B yles operating at te maximum work (MW) output also ave te effiienies as η A and Diesel and Atkinson yles ave MW effiienies very lose to η A [3, 4]. A For atual eat engines, in te eat aborsbing proess te temperature of te working medium sould be lower tan tat of te ot reservoir and in te eat releasing proess te temperature of
te working medium sould be iger tan tat of te old reservoir to estabilis te eat exanging proesses. e eat exanging proesses are no longer isotermal. o disribe te eat exanging between te working meium and te eat reserviors, different eat transfer laws ave been systematially studied [5-7]. e genearal eat transfer law is depited as n n s Q k ( ), were k is te eat ondutane; s and are te temperatures of te eat reservoir and te working medium, respetively. Wen n1, it returns to te linear transfer law[8, 9], and te Stefan-boltzmann radiation law if n4. erefore it is of universality. In addition te Dulong-Petit nonlinear eat transfer law is also investigated [1]. However all te eat transfer laws studied before assume te eat ondutane stays onstant during te eat exanging proess. For atual eat transfer proess, as te development of te eat transfer, te temperature differene between te working medium and te eat reservoir is dereasing. erefore te eat ondutane sould derease as well. e eat ondutane sould not be kept onstant, rater be a timedependent dereasing variable. In tis paper, we assume te eat transfer law between te eat reservoir and te working medium onfirms to te linear law wit time-dependent eat ondutane. e effiieny at te maximum work output is dedued. And te maximum work wi an be extrated is also proposed. As to eat engines, a ertain amount of eatq is absorbed from te ot reservoir ( ), and some of wi Q, is evauated to te old reservoir ( ) at te end of a yle. e eat transfer law between te eat soure and te working medium is assumed to onform to Newton s law of ooling: dq dt d m k( s ) (1) dt were is te eat apaity, m is te working substane mass, is te working substane temperature, is eat soure temperature, k is te eat ondutane. In tis paper, we assume s te eat ondutane between te working medium and te ot and old reservoirs onform to te n n following power-law relations: k a( ), k b( ), n >, were te subsripts ( and ) indiate te ot and old reservoirs; a,b, and are te initial eat ondutane and initial temperatures of te working substane at te beginning of te eating and ooling proesses, respetively. Aording to Eq.(1), te working substane temperature in te eat absorbing proess is a funtion of time t:
( ) ( )( / 1) w t + n Ψ + (2) were Ψ m / a, wi reflets te temperature inrease degree of te working medium in te time absorbing proess and as te dimension of time. e time duration is denoted as, te eat absorbed from te ot reservoir an be alulated as: ( ) ( w )[1 ( / Ψ + 1) ] Q k m n (3) e relative entropy ange of te working substane in te eat absorbing proess is given by: dq + ( )( n / Ψ + 1) s ln m (4) Similarly, te temperature of te working medium, te eat evauated to te old reservoir and te entropy ange during te eat releasing proess are given by 1/n ( ) ( )( / Ψ + 1) w t nt (5) ( ) ( w )[1 ( / Ψ + 1) ] Q k m n (6) s dq ( )( n / Ψ + 1) ln m (7) were Ψ m / b, wi reflets te temperature inrease degree of te working medium in te time releasing proess and as te dimension of time. is te time duration of tat proess. In tis paper, we assume tat te ompressing and expanding proesses are isentropi and te times for ompleting tose proesses are zero. After a yle, te working substane return to its initial state, and te total entropy ange of te working substane sould be zero i.e. s + s. en we ave ( )( n / Ψ + 1) + ( )( n / Ψ + 1) 1 (8) e work extrated during te yle is W Q + Q, and te effiieny isη 1 + Q / Q. ombining Eq.(8) and maximizing W wit respet to, we ave
n n n 1 ( + 1) + 1 η ( + 1) [1 ( + 1) ] Ψ Ψ Ψ n n 1 η{1 [( + 1)( + 1)] } Ψ Ψ (9) Substituting Eq.(9) into Eq.(8), we an also obtain te imal initial temperature of te working medium in te eat releasing proess. n n n η + + + + 1 [1 ( 1) ] ( 1) [1 ( 1) ] Ψ Ψ Ψ n n n 1 ( + 1) + ( + 1) [1 ( + 1) ] Ψ Ψ Ψ (1) From te above equations, we an obtain te effiieny at MWη 1 1η η m A. It is independent of te time duration in eiter proess, and is te A effiieny, wi as been obtained toug and JB yles at MW [3, 4, 11]. Altoug te upper bounds of te effiieny are te same, our model is more universal and general. our model an deribe te J-B and Otto yles if We let, v, respetively. Wat is more, tis model does not speify te p termodynami pates in te eat exanging proess. erefore it an desiribe any arbitraty termodynami pat of te eat exanging proess and is more pratial and realisti tan te traditional ones. We an get te temperature profiles in te eat absorbing and releasing proess by substituting Eqs.(1) into Eq.(2) and substituting Eq. (9)into Eq.(5). e work output an be rewritten as max n n W mη A[1 ( + 1) ][1 ( + 1) ] Ψ Ψ 1 1η 1 ( + 1) + ( + 1) [1 ( + 1) ] 1 [( + 1)( + 1)] Ψ Ψ Ψ Ψ Ψ ( ) n n n n n (11) 2 Eq.(11) aieves its maximum valuew max+ m η wen / Ψ and / Ψ A. It indiates tat even te working medium exanges eat suffiiently wit te eat reservoirs, te max+ work wi ould be extrated is finite and limited by W wi is determined by te eat apaity and te temperatures of ot and old reservoirs. Wile in ideal arnot eat engines, tere is no limitation of work output, oter tan te effiieny. Aording to Eq.(11), we ave te power max output at te maximum work PW /( + ). Altoug wen / Ψ and
/ Ψ, max W aieves it maximum value, but te effiieny stays unanged and is still te A effiieny. However its power output is zero. We define / Ψ as te dimensionless ontat time, wi reflets te equilibrium degree of te temperature between te working medium and eat reservoirs. Aording to Eqs.(9) and(1), wen / Ψ and / Ψ, we ave and 1η. For / Ψ, te eat absorbing proesses are sort enoug so tat te final temperature of te working substane is almost equal to its initial temperature. Furtermore () t to establis te eat w releasing proess. en we ave w() t and () t 1η. It means te eat exanging proesses are isotermal. w Wen / Ψ and / Ψ, we ave / 1η and. For / Ψ, te eat releasing proess is sort enoug so tat te final temperature of te working substane is almost equal to its initial temperature. Furtermore w() t to establis te eat exanging proess. en we ave w() t and () t / 1η, It w means tat te temperatures of te working medium also keep onstant in te eat exanging proesses. But te temperature profiles are not like tose obtained under te situation: / Ψ and / Ψ. For situations were / Ψ and / Ψ, te final temperatures are te same as te initial ones. erefore () t, w() t. e eat ondutane are also kept w onstant and are te initial ones, respetively. However te limits of Eqs.(9) and (1) do not exit. We assume te time durations fulfill te relation / b/ a wi is obtained troug te endoreversible model under te maximum power output [2]. Applying te limit / Ψ, we also dedue te same temperature profiles as tose in Ref. [2]. Meanwile P also aieves its maximum value. Furtermore te orresponding effiieny is also η A.at is to say, te endoreversible model is reovered under tese situations.
In addition, wen / Ψ, / Ψ, we ave / 1η. e eat absorbing proess is long enoug so tat te final temperature of te working substane is almost equal to tat of te eat reservoir, i.e. ( ). And its initial temperature is w n n { 1 η [1 ( + 1) ] + ( + 1) } (12) Ψ Ψ It depends on te ontat time of te old reservoir. For situations were / Ψ, / Ψ, we ave 1η. e eat releasing proess is long enoug so tat te final temperature of te working substane is almost equal to tat of te eat reservoir, i.e. ( ). And its initial temperature is w n ( 1η 1)( + 1) + 1 Ψ (13) 1η It depends on te ontat time of te ot reservoir. Furtermore, wen / Ψ and / Ψ, we ould obtain te initial temperatures, / 1η, 1η, and te final temperatures ( ), ( ) for te eat absorbing and releasing proesses, w w respetively. Representative ases ave been studied to investigate te impat of power index n on te imal eat engine yles as depited in Fig. 1. e imal temperature profile of te working medium in te eat absorbing proess is nearly a linear funtion of S and moves upward wit inreasing n. As we all know, temperature profile sould be onave, and to te most extend an be linear in te -S diagram. erefore, te eat absorbing proess approaes its maximum ability in our imal onditions. Under te same dimensionless ontat time, te lower n leads to a lager eat ondutane, tus more work ould be extrated. e temperature differenes between te eat reservoirs and working medium in bot eat exanging proesses are very signifiant, from wi te irreversible entropy generation stems. In addition, by onsidering various n, we ould ave a furter insigt into oter yles wose temperature profiles during te eat exanging proesses are not onstant su as J-B, Otto yles and any oter yles wit isentropi ompression and expansion proesses.
FIG. 1 e -S diagrams of different imal eat engine yles under different n ( n,.5,1,2 ), were 5K, 3K, / Ψ / Ψ 1 and m 1J / K In onlusion, we ave onduted as analysis of effiieny and its bounds at maximum work output for arnot-like eat engines wose eat transfer proesses are desribed by Newton s law of ooling. But for generality, te eat ondutane are no longer treated as onstants, rater as time-dependent dereasing variables. e upper bound of te effiieny is found to be te A effiieny, and is independent of te time duration ompleting eiter proess or te time-dependent ondutane. Furtermore even te working medium exanges eat suffiiently wit te eat 2 reservoirs, te work wi ould be extrated is finite and limited byw max+ m η. e imal temperature profiles in te eat exanging proesses are analyzed under different dimensionless ontat time limits. Wen te dimensionless ontat times satisfy ertain relations, te endoreversible model is reovered. Furtermore representative ases ave been studied to investigate te effet of n on te imal temperature profiles. e results in present paper ould offer us a furter insigt into any eat engine yles wit isentropi ompression and expansion proesses. is migt be of great guidane for designing or operating atual eat engines. A
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