Research Article Substance Independence of Efficiency of a Class of Heat Engines Undergoing Two Isothermal Processes

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1 hermodynamis olume 0, Artile ID 6797, 5 pages doi:0.55/0/6797 Researh Artile Substane Independene of Effiieny of a Class of Heat Engines Undergoing wo Isothermal roesses Y. Haseli Department of Mehanial Engineering, Eindhoven University of ehnology,.o. Box 5, 5600 MB Eindhoven, he Netherlands Correspondene should be addressed to Y. Haseli, y.haseli@tue.nl Reeived 9 November 00; Aepted Marh 0 Aademi Editor: Y. eles Copyright 0 Y. Haseli. his is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. hree power produing yles have been so far known that inlude two isothermal proesses, namely, Carnot, Stirling, and Erisson. It is well known that the effiieny of the Carnot yle represented by L / H is independent of its working fluid. Using fundamental relationships between thermodynami properties inluding Maxwell s relationships, this paper shows in a losedform thatthe Erisson andthe Stirling yles alsopossess the Carnoteffiieny irrespetive of the nature of the working gas.. Introdution A neessary step when teahing the seond law in undergraduate thermodynamis lasses is to introdue the onept of Carnot yle, whih was invented by Sadi Carnot, a Frenh physiist and military engineer, in 8. Historially, Carnot yle has played a substantial role in further development of the seond law of thermodynamis by Rudolf Clausius, a German physiist and mathematiian, who in 85 showed analytially that using an ideal gas as the working medium, the effiieny of the Carnot yle is / [], where and denote the lowest and the highest temperatures of the yle, respetively. his result led him to invent entropy in 865, and to derive his well-known inequality; that is, (Q/ 0. Later on, these onepts were applied by several otherpioneerssuhasrevor[], Kenrik [], Kleeman [], and Cantelo [5], to establish fundamental ideas. In these works, the Carnot yle was treated as the regime of fully reversible, whereby satisfying (Q/ = 0. Under the assumption of working substane behaving as an ideal gas, the Carnot yle is not the only one whose effiieny beomes /. Erisson and Stirling yles analsoobtainthesameeffiieny [6]. hese two yles are similar to the Carnot yle in a sense that they inlude two isothermal proesses. However, the remaining two proesses in the Erisson yle are isobari while in the Stirling yle they are isohori. he S and diagrams of these yles are depited in Figure. As in Carnot yle, the Erisson and the Stirling yles would also satisfy (Q/ = 0attheregime of fully reversible. Worthy of noting is that Stirling engine was invented by a Sottish engineer, Robert Stirling, in 86; eight years before Carnot published his ideas in a 65-page book. In 8, John Erisson, an Amerian Swedish-born inventor and mehanial engineer, patented the first Erisson yle, whih is nowadays alled Brayton yle. he Erisson yle that we know today, with its idealized S and diagrams shown in Figure, was invented in 85. From a historial point of view, these three yles whih undergo isothermal ompression and expansions proesses, were designed in the same era. Nonetheless, only Carnot yle has historially prevailed in the literature partly due to Benoit Clapeyron s efforts, another Frenh engineer and physiist, who brought Carnot yle to light by desribing it in a graphial presentation in 8, and mainly due to Clausius who formulated the first law and the seond law on the basis of a Carnot yle operating with an ideal gas. As is well-known, the effiieny of Carnot engine is independent of the nature of the working fluid. he easiest proof is to refer to the orresponding S diagram in Figure. he net entropy inrease of the working fluid during the isothermal expansion proess is equal to the net entropy derease of the working medium in the isothermal ompression proess. his implies that Q H / = Q L /

2 hermodynamis Carnot S (a (b Stirling S ( (d Erisson S (e (f Figure : ressure-volume and temperature-entropy diagrams of Carnot, Stirling, and Erisson yles. whih leads to an effiieny of /. he question whih an be naturally asked is whether or not the other two ompeting yles would still have the effiieny / when operating with a working fluid other than the ideal gas; for instane with an der Waals gas. his idea is followed as the main task of the present note; to investigate the effiieny of the idealized Stirling and Erisson engines when operating with an arbitrary working gas. he analysis will be performed with the aid of the fundamental relationships between thermodynami properties. Our starting point will be the following two well-established equations du = ds d, ( dh = ds + d. (

3 hermodynamis hese equations are the onsequene of the first law; that is, du = δq δw. We will also need the following two relationships originally derived by Maxwell ( ( S =, ( ( S ( =. ( he above relationships an be found in any standard thermodynamis textbook; for example, [7, 8].. Stirling Cyle Operating with an Arbitrary Gas he ideal Stirling yle (see Figure onsists of the following four proesses: isothermal ompression whih inludes removal of heat from the system; isohori heat addition; isothermal expansion with heat addition to the system; isohori heat rejetion. o be able to preisely alulate the amount of heat transfer and work in various proesses of the Stirling engine, we need to derive an equation for the internal energy U whih an be expressed as a funtion of temperature and volume; that is, U = U(,. aking into aount the definition of the speifiheatatonstantvolume,thedifferential form of the internal energy an be expressed as du = ( U d + ( U d = d + ( U d. On the other hand, entropy S an be expressed as a funtion of and,too,withadifferential form as ( ( ( ( S S S ds = d + d = d + d. (6 Note that (isemployedin(6. Substituting (6into(, we get ( [ ( ] S du = d + d. (7 It an be implied from (5 and(7 that = d. (8 Hene, (7 and(6 anberewrittenas du = d + [ ( (5 ] d, (9 ds = ( d + d. (0 Applying ( / ( U/ = ( / ( U/ to(9 gives ( ( =. ( Integrating ( at onstant volume leads to a funtion for that is only dependent on temperature. he entropy hange of the working gas during the isohori proesses and an be evaluated using (0 S S = d = F ( F ( F ( = d, S S = d = F ( F ( = F ( F (. ( From ( we onlude that S S = S S or S S = S S ( Applying the first law to the heat addition proess while taking into aount (9, we get Q = du = d ( = F ( F (, where F ( = d. Likewise, the amount of heat rejetion during proess is determined as Q = F ( F ( = F ( F (. (5 Hene, the amount of heat addition during the isohori proess is the same as the amount of heat rejeted during the isohori proess. he amount of heat removed during proess an be used to meet the heat requirement of proess. In a similar manner, we an determine the entropy hange and the amount of heat addition/rejetion during the isothermal proesses and. Using (0 forthe isothermal proesses, we have ( S S = d, ( (6 S S = d. From ( and(6, we onlude that ( ( d = d. (7 he heat addition/rejetion during proess / an be obtained by applying the first law δq = du + d [ = d + = ( d. ( ] d + d (8

4 hermodynamis Hene, ( Q = d, (9 ( Q = d. (0 Realling that the amount of heat addition in proess equals the amount of heat rejetion in proess, the effiieny of the Stirling yle is determined as follows: η S = Q. ( Q Substituting (9 and(0 into( and taking into aount (7, we finally find η S =. (. Erisson Cyle Operating with an Arbitrary Gas he ideal Erisson yle (see Figure inludes the following four proesses: isothermal ompression whih inludes removal of heat from the system; isobari heat addition; isothermal expansion and heat addition to the system; isobari heat rejetion. We need to take the same proedure outlined in the preeding setion to determine the thermal effiieny of the Erisson yle. However, in this setion, we will employ the definition of enthalpy for the analysis of the Erisson yle. For this purpose, suppose that enthalpy H an be expressed as a funtion of temperature and pressure; that is, H = H(,, or in a differential form as dh = ( H d + ( H d = d + ( H d. ( We further suppose that entropy S an be expressed as a funtion of and with a differential form as below in whih ( isemployed. ds = d + d = d ( d. ( Substituting ( into( yields ( [ ( ] S dh = d + d. (5 From ( and(5, we onlude that = d. (6 Hene, [ ( ] dh = d + d. (7 Eliminating dh between (7 and( and rearranging the resulting expression for S gives ds = ( d d. (8 Applying ( / ( H/ = ( / ( H/ to(7 yields ( ( =. (9 Equation (9 impliesthat at an isobari proess is only a funtion of temperature. So, the entropy hange of the working substane during the isobari proesses and an be evaluated using (8 as follows: S S = d = F ( F ( F ( = d, S S = d = F ( F ( = F ( F (. (0 It an be inferred from ( that the entropy inrease during proess is equal to the entropy derease in proess. Hene, S S = S S. ( he amount of heat addition during isobari proess an be obtained using the first law as follows. Q = dh = d = F ( F (, ( where F ( = d. Likewise, the amount of heat rejetion during isobari proess is determined as Q = F ( F ( = F ( F (. ( It an be implied from ( and( thatq = Q.So, the amount of heat rejeted during proess anbe bypassed to meet the heat requirement of proess. he hange in entropy during the isothermal proesses and is evaluated with the aid of (8 ( S S = d, ( ( S S = d. A ombination of ( and( leadsto ( ( d = d. (5 he heat addition/rejetion during proess / an be obtained by applying the first law δq = dh d [ = d + = ( d. ( ] d d (6

5 hermodynamis 5 Hene, ( Q = d, ( Q = d. (7 Finally, taking into aount (5 theeffiieny of the Erisson yle is obtained as follows η E = Q = ( / d Q ( / d (8. Disussion =. he important aspet of the results presented in this note is that the Carnot yle is not the only one whih possesses the highest effiieny among engines operating between the same low- and high-temperature thermal reservoirs. Indeed, the idealized Erisson and Stirling engines are as effiient as the Carnot engine. hese results reveal that the lass of heat engines whih undergo two isothermal proesses has the same effiieny irrespetive of the nature of the working substane. So, the Erisson and the Stirling yles would deserve to be introdued in thermodynamis lassrooms and textbooks; espeially onsidering their pratial importane. he Erisson yle is the limiting design of a gas turbine engine operating on the basis of a Brayton yle with multiple interooler and reheater, whereas the Stirling yle is the ideal design of a reuperating external ombustion engine. Both reversible and irreversible onfigurations of the Erisson and the Stirling engines have reeived tremendous attention in the literature. Among many sholars researhing in these fields, the ontribution of Organ [9, 0] who has been ontinuously working on Stirling engines is aknowledgeable. Referring to the available historial evidenes, one would realize why the Carnot yle, among the group of heat engines with two isothermal proesses, has prevailed in the literature as the most effiient one. his is due to firstly Sadi Carnot s book in whih he attempted to prove by means of rational reasoning (in his era neither the first law nor the seond law had been fully realized and formulated that only his proposed engine would produe a maximum work from a given amount of heat [], even though his ideas were remained unnotied for a deade. On the other hand, Clapeyron played a key role by representing the Carnot s yle graphially in a diagram suh as shown in Figure. More importantly, Carnot s rationalized theorems together with Clapeyron s work allowed Clausius to further advane borders of thermodynamis by formulating the first and the seond laws on the basis of the Carnot yle. Worthy of further disussion is performane of the Erisson and the Stirling engines when exhanging finite time heat with given high and low temperature reservoirs. In 975 Curzon and Ahlborn [] ritiized that a Carnot yle undergoing infinitesimal proesses would physially result in zero power output. hey proposed a model of endoreversible Carnot yle exhanging finite time heat with thermal reservoirs, and showed that the effiieny at maximum power operation obeys η CA = ( L / H 0.5, with L and H representing the temperatures of thermal reservoirs. Either of the Stirling yle or the Erisson yle ould also operate with the above effiieny at maximum power ondition. he key part of the Curzon-Aholborn s demonstration is to employ Q /Q = / (referring to thenomenlatureoffigure. As was shown in the present note, the effiieny of the Stirling and the Erisson yles obeys /. hus, it an be demonstrated with the same methodology of Curzon-Aholborn [] that the effiieny of these yles would be η CA = ( L / H 0.5 at maximum power operation. Aknowledgment he author would like to thank the reviewers for performing a areful review. Referenes [] I. Müller, A History of hermodynamis: he Dotrine of Energy and Entropy, Springer, New York, NY, USA, 007. [] J. E. revor, he exposition of the entropy theory, hysial Chemistry, vol., no. 6, pp. 5 58, 900. [] F. B. Kenrik, A mehanial model to illustrate the gas laws, hysial Chemistry, vol. 8, no. 5, pp. 5 56, 90. [] R. D. Kleeman, he absolute zero of the ontrollable entropy and internal energy of a substane or mixture, hysial Chemistry, vol., no. 5, pp , 97. [5] R. C. Cantelo, he seond law of thermodynamis in hemistry, JournalofhysialChemistry, vol., no. 7, pp , 98. [6] A. M. Y. Razak,Industrial Gas urbines: erformane and Operability, aylor & Franis, Boa Raton, Fla, USA, 00. [7] G. van Wylen, R. Sonntag, and C. Borgnakke, Fundamental of Classial hermodynamis, John Wiley & Sons, New York, NY, USA, 99. [8] A. Bejan, Advaned Engineering hermodynamis, JohnWiley & Sons, New York, NY, USA, 997. [9] A. J. Organ, he Regenerator and Stirling Engine,JohnWiley& Sons, New York, NY, USA, 997. [0] A. J. Organ, he Air Engine: Stirling Cyle ower for a Sustainable Future, CRC ress, 007. [] R. H. hurston, Ed., Refletions on the Motive ower of Heat, John Wiley & Sons, New York, NY, USA, nd edition, 897. [] F. L. Curzon and B. Ahlborn, Effiieny of a arnot engine at maximum power output, Amerian hysis, vol., no., pp., 975.

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