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Hindawi Publishing Corporation Journal of 2011, Article ID 647937, 5 pages doi:10.1155/2011/647937

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

Y. Haseli

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Correspondence should be addressed to Y. Haseli, [email protected]

Received 29 November 2010; Accepted 31 March 2011

Academic Editor: Y. Peles

Copyright © 2011 Y. Haseli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original is properly cited.

Three power producing cycles have been so far known that include two isothermal processes, namely, Carnot, Stirling, and Ericsson. It is well known that the efficiency of the represented by 1 − TL/TH is independent of its working fluid. Using fundamental relationships between thermodynamic properties including Maxwell’s relationships, this paper shows in a closed form that the Ericsson and the Stirling cycles also possess the Carnot efficiency irrespective of the nature of the working gas.

1. Introduction are depicted in Figure 1. As in Carnot cycle, the Ericsson and the Stirling cycles would also satisfy (Q/T) = 0attheregime A necessary step when teaching the second law in undergrad- of fully reversible. uate thermodynamics classes is to introduce the concept of Worthy of noting is that Stirling was invented Carnot cycle, which was invented by Sadi Carnot, a French by a Scottish engineer, Robert Stirling, in 1816; eight years physicist and military engineer, in 1824. Historically, Carnot before Carnot published his ideas in a 65-page book. In cycle has played a substantial role in further development 1833, , an American Swedish-born inventor of the second law of thermodynamics by , a and mechanical engineer, patented the first Ericsson cycle, German physicist and mathematician, who in 1854 showed which is nowadays called . The Ericsson cycle analytically that using an as the working medium, that we know today, with its idealized T–S and P–V the efficiency of the Carnot cycle is 1 − T1/T2 [1], where T1 diagrams shown in Figure 1, was invented in 1853. From a and T2 denote the lowest and the highest temperatures of the historical point of view, these three cycles which undergo cycle, respectively. This result led him to invent “” isothermal compression and expansions processes, were in 1865, and to derive his well-known inequality; that is, designed in the same era. Nonetheless, only Carnot cycle has (Q/T) ≤ 0. Later on, these concepts were applied by several historically prevailed in the literature partly due to Benoit otherpioneerssuchasTrevor[2], Kenrick [3], Kleeman [4], Clapeyron’s efforts, another French engineer and physicist, and Cantelo [5], to establish fundamental ideas. In these who brought Carnot cycle to light by describing it in a works, the Carnot cycle was treated as the regime of fully graphical presentation in 1834, and mainly due to Clausius reversible, whereby satisfying (Q/T) = 0. who formulated the first law and the second law on the basis Under the assumption of working substance behaving of a Carnot cycle operating with an ideal gas. as an ideal gas, the Carnot cycle is not the only one whose As is well-known, the efficiency of Carnot engine is efficiency becomes 1 − T1/T2. Ericsson and Stirling cycles independent of the nature of the working fluid. The easiest canalsoobtainthesameefficiency [6]. These two cycles are proof is to refer to the corresponding T–S diagram in similar to the Carnot cycle in a sense that they include two Figure 1. The net entropy increase of the working fluid isothermal processes. However, the remaining two processes during the isothermal expansion process is equal to the net in the Ericsson cycle are isobaric while in the entropy decrease of the working medium in the isothermal they are isochoric. The T–S and P–V diagrams of these cycles compression process. This implies that QH /T3 = QL/T1 2 Journal of Thermodynamics

P T 3 Carnot Qin Qin 3 4

4

2

Qout 21 1 Qout

V S (a) (b) P T 3 Stirling Qin 3 4 Qin

2

4

21 1 Q Qout out

V S (c) (d) P T 2 3 Ericsson Qin 3 4 Qin

Qout 4 21 1 Qout

V S (e) (f)

Figure 1: -volume and temperature-entropy diagrams of Carnot, Stirling, and Ericsson cycles.

which leads to an efficiency of 1 − T1/T3.Thequestion be performed with the aid of the fundamental relationships which can be naturally asked is whether or not the other two between thermodynamic properties. Our starting point will competing cycles would still have the efficiency 1 − T1/T3 be the following two well-established equations when operating with a working fluid other than the ideal gas; for instance with Van der Waals gas. This idea is followed = − as the main task of the present note; to investigate the dU TdS PdV, (1) efficiency of the idealized Stirling and Ericsson engines when operating with an arbitrary working gas. The analysis will dH = TdS + VdP. (2) Journal of Thermodynamics 3

These equations are the consequence of the first law; that Integrating (11) at constant volume leads to a function for cV is, dU = δQ − δW. We will also need the following two that is only dependent on temperature. relationships originally derived by Maxwell Theentropychangeoftheworkinggasduringthe     isochoric processes 2 → 3and4 → 1 can be evaluated ∂S ∂P = , (3) using (10) ∂V T ∂T V     3 − = cV = − = cV ∂S =− ∂V S3 S2 dT F1(T3) F1(T2) F1(T) dT, . (4) 2 T T ∂P T ∂T P 1 cV The above relationships can be found in any standard S1 − S4 = dT = F1(T1) − F1(T4) = F1(T2) − F1(T3). thermodynamics textbook; for example, [7, 8]. 4 T (12)

2. Stirling Cycle Operating with an From (12) we conclude that Arbitrary Gas S3 − S2 = S4 − S1 or S1 − S2 = S4 − S3 (13) The ideal Stirling cycle (see Figure 1) consists of the following four processes: 1 → 2 isothermal compression which Applying the first law to the heat addition process 2 → 3 includes removal of heat from the system; 2 → 3isochoric while taking into account (9), we get heat addition; 3 → 4 isothermal expansion with heat 3 3 addition to the system; 4 → 1 isochoric heat rejection. To Q23 = dU = cV dT be able to precisely calculate the amount of heat transfer and 2 2 (14) work in various processes of the , we need = F (T ) − F (T ), to derive an equation for the U which can 2 3 2 2 be expressed as a function of temperature and volume; that where F2(T) = cV dT. is, U = U(T, V). Taking into account the definition of the ff Likewise, the amount of heat rejection during process specificheatatconstantvolumecV ,thedi erential form of 4 → 1 is determined as the internal energy can be expressed as       Q = F (T ) − F (T ) = F (T ) − F (T ). (15) ∂U ∂U ∂U 41 2 1 2 4 2 2 2 3 dU = dT + dV = cV dT + dV. ∂T V ∂V T ∂V T Hence, the amount of heat addition during the isochoric (5) process 2 → 3 is the same as the amount of heat rejected → On the other hand, entropy S can be expressed as a function during the 4 1. The amount of heat → of T and V,too,withadifferential form as removed during process 4 1 can be used to meet the heat →         requirement of process 2 3. ∂S ∂S ∂S ∂P In a similar manner, we can determine the entropy dS = dT + dV = dT + dV. ∂T V ∂V T ∂T V ∂T V change and the amount of heat addition/rejection during the (6) isothermal processes 3 → 4and1 → 2. Using (10)forthe isothermal processes, we have Note that (3)isemployedin(6). Substituting (6)into(1), we   get 4 − = ∂P       S4 S3 dV, ∂S ∂P 3 ∂T V = −   (16) dU T dT + T P dV. (7) 2 ∂T V ∂T V ∂P S2 − S1 = dV. It can be implied from (5)and(7)that 1 ∂T V   ∂S From (13)and(16), we conclude that = T cV dT. (8)     ∂T V 4 2 ∂P = ∂P dV dV . (17) Hence, (7)and(6)canberewrittenas 3 ∂T V 1 ∂T V     ∂P The heat addition/rejection during process 3 → 4/1 → 2 dU = cV dT + T − P dV, (9) ∂T V can be obtained by applying the first law   c ∂P dS = V dT + dV. (10) δQ = dU + PdV T ∂T V     ∂P Applying (∂/∂T)(∂U/∂V) = (∂/∂V)(∂U/∂T)to(9)gives = cV dT + T − P dV + PdV ∂T V (18)     2   ∂cV = ∂ P ∂P T 2 . (11) = T dV. ∂V T ∂T V ∂T V 4 Journal of Thermodynamics

Hence, Eliminating dH between (27)and(2) and rearranging the   resulting expression for S gives 2 = ∂P   Q12 T1 dV, (19) cPdT ∂V 1 ∂T V dS = − dP. (28)   T ∂T P 4 ∂P = Q34 = T3 dV. (20) Applying (∂/∂T)(∂H/∂P) (∂/∂P)(∂H/∂T)to(27)yields 3 ∂T V     2 ∂cP ∂ V Recalling that the amount of heat addition in process 2 → 3 =−T . (29) ∂P ∂T2 equals the amount of heat rejection in process 4 → 1, the T P efficiency of the Stirling cycle is determined as follows: Equation (29)impliesthatcP at an is only a function of temperature. So, the entropy change of the Q12 → ηS = 1 − . (21) working substance during the isobaric processes 2 3and Q34 4 → 1 can be evaluated using (28) as follows:

3 Substituting (19)and(20)into(21) and taking into account cP cP S3 − S2 = dT = F3(T3) − F3(T2) F3(T) = dT, (17), we finally find 2 T T

1 = − T1 cP ηS 1 . (22) S1 − S4 = dT = F3(T1) − F3(T4) = F3(T2) − F3(T3). T3 4 T (30) 3. Ericsson Cycle Operating with an It can be inferred from (3) that the entropy increase during Arbitrary Gas process 2 → 3 is equal to the entropy decrease in process 4 → 1. Hence, The ideal Ericsson cycle (see Figure 1) includes the following four processes: 1 → 2 isothermal compression which S1 − S2 = S4 − S3. (31) includes removal of heat from the system; 2 → 3isobaric The amount of heat addition during isobaric process 2 → 3 heat addition; 3 → 4 isothermal expansion and heat → can be obtained using the first law as follows. addition to the system; 4 1 isobaric heat rejection. We 3 3 need to take the same procedure outlined in the preceding = = = − ffi Q23 dH cPdT F4(T3) F4(T2), (32) section to determine the thermal e ciency of the Ericsson 2 2 cycle. However, in this section, we will employ the definition where F (T) = c dT. of for the analysis of the Ericsson cycle. For this 4 P Likewise, the amount of heat rejection during isobaric purpose, suppose that enthalpy H can be expressed as a process 4 → 1 is determined as function of temperature and pressure; that is, H = H(T, P), ff or in a di erential form as Q41 = F4(T1) − F4(T4) = F4(T2) − F4(T3). (33)       ∂H ∂H ∂H It can be implied from (32)and(33)thatQ23 =−Q41.So, dH = dT + dP = cPdT + dV. ∂T P ∂P T ∂V T the amount of heat rejected during process 4 → 1canbe (23) bypassed to meet the heat requirement of process 2 → 3. The change in entropy during the isothermal processes We further suppose that entropy S can be expressed as a 3 → 4and1 → 2 is evaluated with the aid of (28) function of T and P with a differential form as below in   4 which (4)isemployed. ∂V S4 − S3 =− dP,         3 ∂T P ∂S ∂S ∂S ∂V   (34) dS = dT + dP = dT − dP. 2 ∂V ∂T P ∂P T ∂T P ∂T P S2 − S1 =− dP. (24) 1 ∂T P A combination of (31)and(34)leadsto

Substituting (24)into(2)yields     4 2       ∂V = ∂V ∂S ∂V dP dP . (35) dH = T dT + V − T dP. (25) 3 ∂T P 1 ∂T P ∂T P ∂T P The heat addition/rejection during process 3 → 4/1 → 2 From (23)and(25), we conclude that can be obtained by applying the first law   ∂S δQ = dH − VdP T = c dT. (26) ∂T P     P ∂V = cPdT + V − T dP − VdP Hence, ∂T P (36)       ∂V ∂V dH = cPdT + V − T dP. (27) =−T dP. ∂T P ∂T P Journal of Thermodynamics 5

Hence, result in zero power output. They proposed a model of   endoreversible Carnot cycle exchanging finite time heat 2 ∂V ffi Q =−T dP, with thermal reservoirs, and showed that the e ciency at 12 1 0.5 1 ∂T P maximum power operation obeys ηCA = 1 − (TL/TH ) ,   (37) 4 ∂V with TL and TH representing the temperatures of thermal Q34 =−T3 dP. reservoirs. Either of the Stirling cycle or the Ericsson cycle ∂T 3 P could also operate with the above efficiency at maximum Finally, taking into account (35)theefficiency of the Ericsson power condition. The key part of the Curzon-Aholborn’s cycle is obtained as follows demonstration is to employ Q12/Q34 = T1/T3 (referring to thenomenclatureofFigure 1). As was shown in the present − 2 ffi Q12 T1 1 (∂V/∂T)PdP note, the e ciency of the Stirling and the Ericsson cycles ηE = 1 − = 1 − − 4 obeys 1 − T1/T3. Thus, it can be demonstrated with the same Q34 T3 (∂V/∂T)PdP 3 (38) methodology of Curzon-Aholborn [12] that the efficiency of T = − 0.5 = 1 − 1 . these cycles would be ηCA 1 (TL/TH ) at maximum T3 power operation.

4. Discussion Acknowledgment The important aspect of the results presented in this note The author would like to thank the reviewers for performing is that the Carnot cycle is not the only one which possesses a careful review. the highest efficiency among engines operating between the same low- and high-temperature thermal reservoirs. Indeed, the idealized Ericsson and Stirling engines are as efficient References as the Carnot engine. These results reveal that the class of [1] I. Muller,¨ A History of Thermodynamics: The Doctrine of Energy heat engines which undergo two isothermal processes has ffi and Entropy, Springer, New York, NY, USA, 2007. the same e ciency irrespective of the nature of the working [2] J. E. Trevor, “The exposition of the entropy theory,” Journal of substance. So, the Ericsson and the Stirling cycles would Physical Chemistry, vol. 4, no. 6, pp. 514–528, 1900. deserve to be introduced in thermodynamics classrooms and [3] F. B. Kenrick, “A mechanical model to illustrate the ,” textbooks; especially considering their practical importance. Journal of Physical Chemistry, vol. 8, no. 5, pp. 351–356, 1904. The Ericsson cycle is the limiting design of a [4] R. D. Kleeman, “The absolute zero of the controllable entropy engine operating on the basis of a Brayton cycle with multiple and internal energy of a substance or mixture,” Journal of and reheater, whereas the Stirling cycle is the ideal Physical Chemistry, vol. 31, no. 5, pp. 747–756, 1927. design of a recuperating external combustion engine. Both [5] R. C. Cantelo, “The second law of thermodynamics in reversible and irreversible configurations of the Ericsson and chemistry,” JournalofPhysicalChemistry,vol.32,no.7,pp. the Stirling engines have received tremendous attention in 982–989, 1928. [6]A.M.Y.Razak,Industrial Gas Turbines: Performance and the literature. Among many scholars researching in these Operability, Taylor & Francis, Boca Raton, Fla, USA, 2003. fields, the contribution of Organ [9, 10] who has been con- [7] G. van Wylen, R. Sonntag, and C. Borgnakke, Fundamental of tinuously working on Stirling engines is acknowledgeable. Classical Thermodynamics, John Wiley & Sons, New York, NY, Referring to the available historical evidences, one would USA, 1994. realize why the Carnot cycle, among the group of heat [8] A. Bejan, Advanced Engineering Thermodynamics,JohnWiley engines with two isothermal processes, has prevailed in the & Sons, New York, NY, USA, 1997. literature as the most efficient one. This is due to firstly Sadi [9] A. J. Organ, The Regenerator and Stirling Engine,JohnWiley& Carnot’s book in which he attempted to prove by means Sons, New York, NY, USA, 1997. of rational reasoning (in his era neither the first law nor [10] A. J. Organ, The Air Engine: Stirling Cycle Power for a the second law had been fully realized and formulated) that Sustainable Future, CRC Press, 2007. only his proposed engine would produce a maximum work [11] R. H. Thurston, Ed., Reflections on the Motive Power of Heat, John Wiley & Sons, New York, NY, USA, 2nd edition, 1897. from a given amount of heat [11], even though his ideas [12] F. L. Curzon and B. Ahlborn, “Efficiency of a carnot engine at were remained unnoticed for a decade. On the other hand, maximum power output,” American Journal of Physics, vol. 43, Clapeyron played a key role by representing the Carnot’s no. 1, pp. 22–24, 1975. cycle graphically in a P–V diagram such as shown in Figure 1. More importantly, Carnot’s rationalized theorems together with Clapeyron’s work allowed Clausius to further advance borders of thermodynamics by formulating the first and the second laws on the basis of the Carnot cycle. Worthy of further discussion is performance of the Ericsson and the Stirling engines when exchanging finite time heat with given high and low temperature reservoirs. In 1975 Curzon and Ahlborn [12] criticized that a Carnot cycle undergoing infinitesimal processes would physically Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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