sign in sign up
<<
Home , Carnot heat engine, Disgregation, Heat engine, Irreversible process, Quasistatic process, Thermodynamic cycle

Reversible and irreversible and cycles Harvey S. Leff

Citation: American Journal of Physics 86, 344 (2018); doi: 10.1119/1.5020985 View online: https://doi.org/10.1119/1.5020985 View Table of Contents: http://aapt.scitation.org/toc/ajp/86/5 Published by the American Association of Physics Teachers Reversible and irreversible and refrigerator cycles Harvey S. Leffa) California State Polytechnic University, Pomona, Pomona, CA 91768 and Reed College, Portland, OR 97202 (Received 26 June 2017; accepted 26 December 2017) Although no reversible thermodynamic cycles exist in nature, nearly all cycles covered in textbooks are reversible. This is a review, clarification, and extension of results and concepts for quasistatic, reversible and irreversible processes and cycles, intended primarily for teachers and students. Distinctions between the latter process types are explained, with emphasis on clockwise (CW) and counterclockwise (CCW) cycles. Specific examples of each are examined, including Carnot, and Stirling cycles. For the , potentially useful task-specific efficiency measures are proposed and illustrated. Whether a cycle behaves as a traditional refrigerator or heat engine can depend on whether it is reversible or irreversible. Reversible and irreversible-quasistatic CW cycles both satisfy Carnot’s inequality for thermal efficiency, g g . Irreversible CCW cycles with two reservoirs satisfy the coefficient of performance  Carnot inequality K KCarnot. However, an arbitrary reversible cycle satisfies K KCarnot when compared with a reversible operating between its maximum and minimum , a potentially counterintuitive result. VC 2018 American Association of Physics Teachers. https://doi.org/10.1119/1.5020985

I. INTRODUCTION contains related final remarks. Throughout the manuscript, main findings are expressed in “Key Points.” Historically, advances in were closely linked to the study of thermodynamic cycles and the limita- II. DISTINCTION BETWEEN REVERSIBLE AND tions on their efficiency for performing , cooling or IRREVERSIBLE CYCLES heating. This was of interest to engineers and physicists ini- tially for clockwise heat engine cycles in connection with Rudolf Clausius2,3 used the distinction between reversible efficiency limits imposed by the second law of thermody- and irreversible CW cycles in his tour de force derivation of namics. Ultimately both clockwise (CW) and counterclock- the so-called Clausius inequality. Using cycles as a mathe- wise (CCW) thermodynamic cycles were studied and both matical tool rather than a device for doing work, heating or are covered in introductory physics textbooks. cooling, Clausius was led to the concept and definition of The question of whether all CCW cycles are , including the principle of entropy increase. was addressed by Dickerson and Mottmann.1 That question Specifically, he examined a CW cycle on a vs. vol- can be posed for heat as well. The answers depend ume diagram such as that in Fig. 1. Define d-Q to be an inex- critically not only on the definitions of a refrigerator and act differential that represents the added to the heat engine (which are to some extent arbitrary) but also on working substance from a reservoir at T; i.e., the important physics-related issue of whether a cycle is d-Q > 0 when the working substance receives energy and reversible or irreversible. This article is intended to clarify d-Q < 0 when the working substance sends energy to the res- these points for physics teachers and students. Additionally ervoir. Clausius showed that for a cyclic process of the work- it is a review, clarification and elaboration, with some new ing substance,4 results for reversible and quasistatic-irreversible processes. d-Q Because it is part a review, some standard textbook material 0; the Clausius inequality: (1) is revisited. To my knowledge, no existing source synthe- T  sizes these concepts, which are fundamental in physics þ teaching. The inequality holds for an irreversible cycle and the Section II is devoted to important distinctions between equality holds for a reversible cycle. Clausius assumed that a reversibility and irreversibility, and Sec. III clarifies charac- (possibly infinite) number of reservoirs guide the system teristics of reversible and irreversible work and heat pro- through the cycle. During the cycle, the working substance cesses. Distinctions between quasistatic and reversible can deviate from thermodynamic equilibrium, but by defini- processes are explained and a pulsed model for quasistatic- tion the reservoir cannot. Figure 1 assumes the cycle can be irreversible heat processes is proposed. Section IV is a brief split into two segments, one irreversible, the other reversible, review of heat engines and refrigerators and their efficiency and Eq. (1) can be rewritten d-Q=T irrev 1 2 þ rev 2 1 measures. Inclusion of temperature vs. entropy graphs is to d-Q=T 0. Reversing the sense of theð ! integralÞ alongð ! theÞ show the temperature behavior along each cycle and also to  Ð Ð reversible path, the latter equation becomes irrev 1 2 reveal heat quantities as areas under curves. - - ð ! Þ dQ=T rev 1 2 dQ=T 0. The second term is defined to Section V is devoted to Kelvin three-legged cycles, Sec. À ð ! Þ  Ð be DS S 2 S 1 and thus, VI covers the well known Carnot cycles, and Sec. VII  Ð ð ÞÀ ð Þ focuses on Stirling cycles. Sections VIII and IX are dedi- cated to thermal efficiency and coefficient of performance DS d-Q=T: (2)  irrev 1 2 inequalities for reversible and irreversible cycles. Section X ð ð ! Þ

344 Am. J. Phys. 86 (5), May 2018 http://aapt.org/ajp VC 2018 American Association of Physics Teachers 344 Key Point 3. A quasistatic path is a continuum of equilib- rium states. An infinite number of reservoirs is needed to take a working substance through paths with variable- temperature heat processes. Real paths must deviate from equilibrium states, and can only approximate quasistatic work and heat processes. Key Point 4. When taken from a to state b quasistatically, a system can traverse those states in b reverse order, and its entropy change is DS d-Q=T ab ¼ a DSba. A is reversible only if it can ¼Àbe run backwards such that both the system andÐ environ- Fig. 1. Clockwise cycle with an irreversible segment 1 2 and a reversible ment follow their same paths in reverse order. All reversible segment 2 1. The dashed line 1 2 is purely symbolic! because pressure processes are quasistatic but a quasistatic process need not P might not! be well defined along this! path, so the path cannot be graphed. be reversible. Key Point 5. It takes a reversible heat engine an infinite amount of to complete each cycle and thus, the The equality holds only if the irreversible path is replaced by output of such a cycle is zero. Similarly, a reversible refrig- a reversible one. Indeed, for a reversible cycle if the inequal- erator would have a cooling rate of zero. The same is true ity held, then reversing the cycle would lead to the opposite for quasistatic cycles. inequality—thus only the zero value is mathematically possi- Despite the practical limitations of reversible cycles, their ble. Reversibility and irreversibility are clarified further in ease of treatment and use in establishing profound efficiency Sec. III. limits are important. Actual heat engines and refrigerators For an irreversible adiabatic path, - 0 and Eq. (2) dQ have unavoidable sources of irreversibility like and implies ¼ heat leaks. They typically operate so fast that their working DS 0 for adiabatically insulated or isolated systems: substances, e.g., , do not have uniform temperatures  and . Nonequilibrium states are involved and such (3) 6 processes are irreversible. Note that a pure work process cannot decrease the entropy of an adiabatically insulated system. The following points are III. CHARACTERISTICS OF REVERSIBLE AND worth emphasizing: IRREVERSIBLE PROCESSES Key Point 1. Equation (3) is a form of the second law of thermodynamics, namely, the principle of entropy increase. A. Finite-speed work processes For an isolated “universe,” DS 0. universe  Thermodynamic processes depend critically on process Key Point 2. It is essential to carefully distinguish between speed. For sufficiently slow processes, the path of thermody- reversible and irreversible processes when examining ther- namic states followed can be approximated using equilib- modynamic cycles. Non-equilibrium states, for which tem- rium states and quasistatic processes, which can be perature and other variables are not well defined, can represented on a pressure- or temperature-volume occur for irreversible processes. In Clausius’s arguments graph such as the lower path 2 1 in Fig. 1. The upper path (2) leading to Eq. , T refers to the temperatures of the 1 2 cannot be graphed because! the irreversibility leads to constant-temperature reservoirs that generate the cycle. By nonequilibrium! states without well defined pressures and/or definition, reservoirs have infinitesimally small relaxation . and are always in equilibrium. For an example, consider an external force on a piston, No reversible processes exist in the real world. If one which induces external pressure Pext on a gas whose equilib- existed, it would take an infinite amount of time because rium pressure is initially P. If Pext P, the gas molecules approaching reversibility in a laboratory requires an ultra- bunch up near the quickly moving piston, and gas pressure slow process with minimal deviations from equilibrium, just near the piston exceeds that farther away; i.e., there is a non- enough to make the process proceed. This entails near equi- equilibrium situation without a well defined pressure, as in librium states, which lie outside the domain of classical ther- Fig. 2(a). If Pext P, the piston moves in the opposite direc- modynamics, and which are arbitrarily close to the tion, and the gas pressure near the piston is less than that equilibrium states on the reversible path. farther away, as in Fig. 2(b). If the receding piston moves 5 John Norton summed up the issue neatly as follows: “…a faster than any of the gas molecules, the gas expands freely reversible process is, loosely speaking, one whose driving and a P-V plot is not possible. forces are so delicately balanced around equilibrium that only a very slight disturbance to them can lead the process to Key Point 6. Only if the external pressure differs infinitesi- reverse direction. Because such processes are arbitrarily mally from P, will a volume change approach the quasi- close to a perfect balance of driving forces, they proceed static ideal, and an infinitesimal external pressure change arbitrarily slowly while their states remain arbitrarily close can reverse the path. If the work sources (e.g., a weight to equilibrium states. They can never become equilibrium hanging from a pulley) and reservoirs also retrace their states, for otherwise there would be no imbalance of driving paths, the process is reversible. Clearly, a quick compres- forces, no change, and no process. Equilibrium states remain sion followed by a quick expansion does not lead to the as they are.” initial gas state.

345 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 345 using a convenient reversible path that connects the system’s initial and final states. It is possible to transfer energy from higher to lower temperature quasistatically, but not reversibly, as shown in Fig. 4. Connect the cycle’s working substance, with tempera- ture reservoirs using ultra-thin low-conductivity fibers. If the connection is closed, opened, and closed again at quick intervals, this creates arbitrarily small, irreversible energy transfers by thermal conduction. Repeating this procedure generates a quasistatic, irreversible heat process. Although pulsing through narrow fibers provides a pleasing mental Fig. 2. Symbolic view of an irreversible quick (a) compression and (b) model, in fact pulsing through larger surfaces can achieve expansion of a gas. the same result. When the working substance undergoes a cyclic process, B. Finite-speed heat processes its entropy change per cycle is zero. Because the pulsed energy transfers between the working substance and reser- For a reversible adiabatic (zero heat) process, the work voirs are through finite temperature differences, there is a net process involved induces temperature changes of a gas with- out reservoirs. More generally, there can be temperature var- entropy gain by the reservoirs during each cycle—i.e., out- iations associated with a combination of work and heat side the working substance—consistent with the second law processes. To approach reversibility, such temperature of thermodynamics. changes must be extremely slow. It is helpful to imagine suc- Key Point 8. Quasistatic-irreversible cycles with two reser- cessive, constant-temperature energy reservoirs, as illustrated voirs can be effected via ultra slow work and pulsed heat in Fig. 3(a). In principle, the limiting case of an infinite num- processes. Reversal using only the same reservoirs is ber of reservoirs with vanishing successive temperature dif- 7 impossible. ferences achieves reversibility. As successive temperature Key Point 9. A variable temperature path can be achieved differences get smaller, the entropy produced does too, for a quasistatic irreversible cycle with two reservoirs, as in approaching zero in the reversible limit. The energy trans- Fig. 4, using an ultra-slow heat process together with a ferred to the system is Q Tn C dT, where C is the sys- sys T1 p p simultaneous slow work process. tem’s constant-pressure heat¼ capacity at an assumed constant Ð atmospheric pressure. The system’s entropy gain DSsys There are other ways a quasistatic- can Tn ¼ occur. One is by friction between a slowly moving piston T Cp=T dT is the negative of that for the reservoirs. 1 ð Þ and its cylinder wall, which converts of ÐKey Point 7. Reversible heating can be done with a the moving piston to of the piston and cylin- sequence of reservoirs, each with an infinitesimally higher der. If the piston’s motion is reversed, again ultra-slowly, . temperature than the last and reversal is possible friction converts more mechanical to internal energy. Such If instead, heating or cooling results from an energy trans- processes dissipate energy and are not reversible. fer through a finite temperature difference such as Tn T1 0 in Fig. 3(b), the process is . The energyÀ > irreversible C. Quasistatic volume changes transfer to the system and its entropy change are the same as in the previous paragraph. Thus the total entropy change is In this section, I address isothermal, isobaric and adiabatic positive which signals irreversibility, quasistatic volume change. Although a heat process unac- companied by a work process requires a temperature gradi- DStotal DSsys DSres ent, an isothermal volume change has no such gradient. ¼ þ T n 1 1 Key Point 10. An ultra-slow volume change of a gas in con- Cp dT > 0 for T1 Tn: (4) ¼ T1 T À Tn  tact with a reservoir generates an ð  5 through a sequence of near equilibrium states. Throughout Although this process is irreversible, the calculation of DSsys the process energy is exchanged between the gas and reser- takes advantage of the property of entropy, voir, keeping the gas temperature constant.

Fig. 4. Depiction of an irreversible cyclic process with a working substance (circle) using ultra-slow, pulsed heat processes through narrow low- Fig. 3. (a) Heating by thermal contact with a sequence of n constant- conductivity fibers connected to hot T and cold T reservoirs. Either ð þÞ ð ÀÞ temperature energy reservoirs, with Tj T1 j; j 0; 1… n 1 . positive or negative work can be done on the working substance, which goes Reversibility is achieved as  0 and n ¼with nþ fixed.¼ (b) Irreversibleð À Þ through a quasistatic cycle, with zero entropy change of the working sub- energy transfer, bringing the system! directly!1 from the first to nth reservoir. stance, but non-zero net entropy increase of the reservoirs.

346 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 346 Key Point 11. A quasistatic process at constant pressure W Qout g 1 : (5) can be induced using a succession of constant-temperature  Qin ¼ À Qin reservoirs. This results in a combination of work and heat processes that redistribute energy between the system and Similarly, a refrigerator cycle’s purpose is to remove reservoirs, leading to a desired final volume or temperature. energy from a lower temperature region, and it is common to Key Point 12. A reversible, adiabatic ultra-slow work pro- define a coefficient of performance K for such CCW cycles, cess on an insulated system alters its internal energy and Q 1 temperature at constant entropy. Examples are (i) a volume K in : (6) change resulting from ultra-slow, variable-pressure piston  W ¼ Qout=Qin 1 work and (ii) magnetization or demagnetization of a para- À magnet slowly and adiabatically, altering its temperature. The general efficiency measures, g and K, exist for any CW Key Point 13. Adiabatic work processes are central to ther- and CCW cycles, but are of most practical interest for modynamics, from connecting higher and lower tempera- “traditional” model heat engines or refrigerators defined ture regions of heat engines and refrigerators to reaching above. ultra-low temperatures not reachable otherwise. For a reversible cycle, Qin in g equals Qout in K and vice versa, and Eqs. (5) and (6) together imply 1 g IV. HEAT ENGINES, REFRIGERATORS, AND K À reversible cycle : (7) EFFICIENCY MEASURES ¼ g ðÞ The focus here is on quasistatic cycles with gaseous work- In Secs. V–VII, I use the Kelvin, Carnot and Stirling cycles ing substances that can be represented on pressure-volume to illustrate some implications of reversibility and irrevers- and temperature-entropy graphs. ibility for CW and CCW cycles. What is a heat engine? A general definition of a heat engine is a cyclic device that takes a working substance V. KELVIN CYCLE through a CW , receiving energy Qin from a range of high temperature reservoirs, delivering positive A. CCW Kelvin cycle work energy W to an external load and the remaining energy One of the earliest studies of CCW cycles was by William Q Q W to a range of low temperature reservoirs. A out in Thomson (Lord Kelvin) who was interested in a novel way special¼ caseÀ is a “traditional” heat engine with narrow high to heat buildings by extracting energy from the outdoor envi- and low temperature ranges relative to the temperature differ- ronment, thereby in essence inventing the concept of the ence between those ranges. An example is an electric generat- .8,9 His idea, modeled (roughly) in Figs. 5(a) and ing plant, where the high temperature range is that of the 5(b), was to take outdoor air into a conducting cylinder burner and the low temperature range is that of the outdoor located outdoors, and to expand that air isothermally at the environment. winter outdoor temperature T (path ab), reducing its pres- What is a refrigerator? A general definition of a refriger- sure. The second step was anÀ adiabatic compression (path ator is a cyclic device that takes a working substance through ) in a now insulated cylinder, raising the gas temperature a CCW thermodynamic cycle, receiving positive external bc from T to T > T . work energy W and heat energy Q from a range of low tem- in Kelvin’sÀ proposedþ À machine did not execute a closed cycle; peratures, delivering energy Q Q W to a range of out in i.e., one with zero exchange of working substance with the higher temperatures. The special¼ caseþ of a “traditional” environment. Rather, he specified that the warmed air in state refrigerator has narrow high and low temperature ranges c was to be transferred at constant pressure to the space relative to the temperature difference between those ranges. being heated, with an equal amount of somewhat cooler An example is a household refrigerator, where the high tem- air being transferred from that space to the outdoors. perature range is that of the kitchen and the low temperature Simultaneously, new, lower-temperature air in state a was to range is that of the food compartment. be taken into the Kelvin warming machine from the out- Q and Q represent net energy input and output in out doors, and the latter set of steps was to be repeated ad magnitudes for the working substance during a cycle. . Subscripts and are reserved for the highest and lowest infinitum The closed cycle in Fig. 5,whichisdubbedthe cycle temperatures.À þ The irreversible-quasistatic cycles “Kelvin cycle,” approximates Kelvin’s proposed machine considered in the remainder of this article are limited to irreversibilities achievable using pulsed energy exchanges with reservoirs. Efficiency measures. Applying the first law of thermody- namics to one complete CW cycle, DU Qin Qout W 0, and the work done by the working substance¼ À is W ÀQ ¼ ¼ in Qout > 0. Doing the same for one complete CCW cycle, theÀ latter equation holds with W being the external work on the working substance. Because the objective of a heat engine is to use energy from a high-temperature region to perform external work, it Fig. 5. (a) Pressure vs. volume for a Kelvin Cycle. (b) Temperature vs. is common to compare CW cycles according to their thermal entropy for the same cycle. Work on the system (CCW cycle) or by the sys- efficiency g tem (CW cycle) is the shaded area abca.

347 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 347 with the quasistatic paths ab and bc, plus a quasistatic, Application of Eq. (6), intended for a multi-reservoir sys- isobaric path leading from state c back to a. This captures tem, gives a meaningless result. the spirit, but not the letter, of Kelvin’s warming machine proposal. B. Clockwise Kelvin cycle 1. Reversible CCW Kelvin cycle Although William Thomson did not consider a CW cycle, one has been studied.10 There are stark differences between The reversible CCW Kelvin cycle abca in Fig. 5 uses an its reversible and irreversible cases. infinite number of reservoirs along ca, covering the tempera- ture interval T ; T . ð À þÞ 1. Reversible CW Kelvin cycle . Key Point 14 For the reversible CCW Kelvin cycle, The constant-pressure expansion ac of the reversible energy Qin Q is taken in at temperature T and Qout is Kelvin cycle’s working substance requires energy inputs at rejected over¼ theÀ interval T ; T ,whichincludesTÀ .The ð À þÞ À all temperatures in T ; T . Figure 6(b) shows that Q absorption and ejection temperature regions are not sepa- Kelvin Carnot ð ÀKelvinþÞ Carnot À ¼ Q Q and Qp < Q , which with Eq. (5) rated. Although this is not a traditional refrigerator, it confirmsÀ ¼ Carnot’sÀ inequality, whichþ is generalized in Sec. does refrigerate, removing net energy from the reservoir VIII, at T in Fig. 5. À Figure 6 shows the reversible Kelvin cycle embedded in a gKelvin < gCarnot: (9) graph of a reversible Carnot cycle with isothermal segments at T and T alternated with adiabatic segments. That figure Key Point 16. The reversible CW Kelvin cycle has some þ À shows that for CCW cycles, the same heat input Qin Q characteristics of a heat engine: most of QKelvin is input at ¼ À p occurs at T for the Kelvin and Carnot cycles. The Kelvin the higher temperatures, output is to the lowest temperature, À cycle’s highest and lowest reservoir temperatures are the and the cycle does external work. Because energy absorp- Carnot cycle temperatures and the Carnot cycle’s adiabatic tion occurs over the full temperature range T ; T , this is ½ À þŠ segments contain states a and c, so the external work WKelvin not a “traditional” heat engine. for the Kelvin cycle (area abca) is less than WCarnot (shaded plus hatched areas, abcda). Equation (6), KCarnot Q =W , gives us 2. Irreversible CW Kelvin cycle  in Carnot For the irreversible CW Kelvin cycle acba, a high temper- KKelvin > KCarnot when Carnot cycle temperatures ature reservoir at T is needed to execute path ac with are T ;T : 8 ultra-slow pulsed heatingþ of the gas from T to T . A low À þ ð Þ temperature (T ) reservoir is needed for theÀ isothermalþ path Figure 6(b) makes clear that the magnitude QCarnot ba. Energy is receivedÀ from the T reservoir and is delivered c Kelvin c þ þ d T dS > Qp a T dS, which also leads to Eq. (8), to the T reservoir. using¼ Eq. (6). Equation¼ (8) is generalized in Sec. IX A. À Ð Ð Key Point 17. The irreversible CW Kelvin cycle with two reservoirs is a traditional heat engine as described in 2. Irreversible CCW Kelvin cycle Sec. IV. No reservoir is needed at T because that temperature is þ To see an explicit indicator of irreversibility, suppose the reached via adiabatic compression. Only the T reservoir is working substance is a monatomic classical . The needed to facilitate the isothermal segment abÀ. Along iso- energy transferred is Qp 5=2 Nk T T and along cb baric path ca, a pulsed heat process sends energy from the the gas does work on the¼ð piston,Þ loweringð þ À itsÀÞ temperature to working substance to the T reservoir. T . The compression ba delivers energy Q NkT ln À À À ¼ À Key Point 15. With only one reservoir, the irreversible Vb=Va to the low temperature reservoir. Along ac, Va=Ta ð Þ 2=3 2=3 CCW Kelvin cycle is not a refrigerator. It removes Q from Vc=Tc and along cb, Vb Tb Vc Tc so Vb=Va À ¼ T =T 5=2, and Q 5=2 NkT¼ ln T =T : The the lone reservoir and adds Qin Q WKelvin to the same ¼ À þ entropy¼ð þ changeÀÞ of the gasÀ per¼ð cycleÞ is ÀDSð þ 0.ÀÞ Because reservoir, generating entropy DS WKelvin=T each cycle. ws ¼ ¼ À Ta Tb T and Tc T , the net entropy change of the gas¼ plus¼ reservoirsÀ is11¼ þ

Qp Q 5 T T DS À Nk 1 À ln À > 0 total ¼ÀT þ T ¼À2 ÀT À T þ À þ þ withT < T : 10 À þ ð Þ More entropy is gained by the low temperature reservoir than is lost from the high temperature reservoir, leading to a net entropy increase, signaling irreversibility and consistency with the second law of thermodynamics. Fig. 6. Kelvin cycle abca, embedded in a reversible Carnot cycle. Note that Notably, the irreversibility does not reduce the thermal Kelvin efficiency relative to the reversible Kelvin cycle Qp Qout; Qin for {CCW, CW} cycles, respectively. Subscript p con- gKelvin notes a¼ constant-pressuref g path. because both cycles have the same Q and W energy transfers.

348 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 348 Generally, the connections between entropy increase and thermal efficiency can be complex.12

VI. CARNOT CYCLES A. Clockwise Carnot cycle The familiar CW Carnot cycle for a classical ideal gas working substance is illustrated on the pressure vs. volume and temperature vs. entropy graphs in Fig. 7.

1. Reversible CW Carnot cycle The reversible Carnot cycle holds a unique role in thermo- Fig. 8. Carnot cycle with (a) rapid, irreversible expansion starting in state b, dynamics. It has two constant-temperature reservoirs, and leading to c0 rather than c; and (b) a rapid compression starting in state d, temperature changes occur only along the adiabatic seg- leading to a0 rather than a. The dashed lines in (a) and (b) are symbolic, rep- resenting paths that cannot be graphed because pressure is not well defined ments. The work done by the working substance is the between states b and c0 and states d and a0. The latter four states are well shaded area of the cycle abcda in Figs. 6(a) and 6(b). This defined. follows from dU TdS pdV 0 for one cycle, implying ¼ À ¼ Þ Þ Þ leads to the modified path da0 in Fig. 8(b). This results in Q0 < Q and g < gCarnot again. W P dV T dS þ þ ¼ abcda ¼ abcda Key Point 19. Internal irreversibilities from rapid volume þ þ changes lower the efficiency of a rela- positive area bounded by abcda: (11) ¼ tive to the corresponding reversible Carnot cycle. Type b irreversibility is illustrated by the cycle, shown Key Point 18. The reversible CW Carnot cycle in Fig. 6 is in Fig. 9ð, thatÞ was introduced to physics teachers by Curzon a traditional heat engine as described in Sec. IV. and Ahlborn in 1975.13 The cycle is sometimes referred to as a Novikov-Curzon-Ahlborn cycle.14,15 It consists of a revers- 2. Irreversible CW Carnot cycle ible Carnot cycle with irreversible energy exchanges with reservoirs at temperatures TH > TL as shown in Fig. 9. As Consider two possible sources of irreversibility: a a pro- normally implemented, this model is internally inconsistent cess that takes the working substance out of equilibrium,ð Þ in that the quasistatic Carnot cycle takes infinite time per e.g., by a rapid, irreversible adiabatic (Q 0) expansion; and cycle while the heat processes take finite times. As an b a quasistatic heat process through a¼ finite temperature approximation, one can envisage long but finite times for all differenceð Þ using the pulsed energy technique in Fig. 4. processes, depicted in Fig. 9, which limits the output power. Type a irreversibility is exemplified by a working For fixed reservoir temperatures TH and TL, there are two ð Þ substance that undergoes a quick adiabatic expansion as in limiting cases in Fig. 9(a). One is T TH and T TL, 12 þ ! À ! Fig. 2(b), replacing segment bc with bc0 in Fig. 8(a). the reversible Carnot limit with zero power output. The other Because the piston moves away from the gas, less work is is, T T , where the work per cycle and power output þ ! À done on it per unit of volume expansion than for the revers- approach zero. For intermediate values of T ; T there is a 13ð À þÞ ible case. In order for the working substance to lose suffi- power maximum with thermal efficiency cient internal energy to lower its temperature to T , the piston must move farther. This results in a subsequentÀ W TL gà 1 < gCarnot isothermal compression c0d with larger output Q0 > Q  Qin ¼ À TH À À rffiffiffiffiffiffi relative to the reversible case. According to the definition in at maximum power output: (12) Eq. (5), the thermal efficiency is g 1 Q0 =Q < gCarnot. In similar fashion, a quick compression,¼ depictedÀ À inþ Fig. 2(b),

Fig. 7. Carnot cycle (CW or CCW) on (a) pressure–volume (P-V) and (b) tem- Fig. 9. Temperature-entropy plot of an irreversible-quasistatic Carnot CW perature–entropy (T-S) plots for a gas working substance. Paths ab and cd are (heat engine) cycle with two isothermal and two adiabatic segments. (b) isothermal while bc and da are adiabatic. The areas abcda in (a) and (b) equal Temperature-entropy plot of the irreversible CCW (refrigerator) cycle. the work magnitude by or on the gas. The shaded hatched; hatched areas Note: T ; T are the working substance’s (maximum, minimum) tempera- ð þ Þ ð þ ÀÞ in (b) equal Q ; Q . tures while TH; TL are the high and low reservoir temperatures. ð þ ÀÞ ð Þ

349 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 349 The Novikov-Curzon-Ahlborn model shares a common fea- 1 K ture with real heat engines: there are finite temperature irrCarnot ¼ T =T 1 differences between the working substance’s highest and ðÞþ À À for an irreversible–quasistatic CCW Carnot cycle: lowest temperatures and the respective high and low reser- voir temperatures. A (non-Carnot) example is a (13) used to generate in a power plant.16 The boiler’s flame temperature exceeds the steam temperature and the Key Point 23. The irreversible CCW Carnot cycle is a tra- water temperature in the condenser exceeds the ambient tem- ditional refrigerator as described in Sec. IV. For a given perature. It turns out that the efficiency gà in Eq. (12) gives Qout, Fig. 9(b) shows the minimum required input work good numerical approximations (likely fortuitous) for elec- occurs in the reversible Carnot cycle limit, and Eq. (6) tric power plants. The efficiency gà occurs also under maxi- implies KirrCarnot KCarnot. mum work (not power) conditions for a number of common  reversible heat engine models.17 VII. STIRLING CYCLE Key Point 20. The irreversible Carnot cycle in Fig. 9(a) is a “traditional” heat engine and obeys Carnot’s inequality, The Stirling cycle has two isothermal segments at temper- g < g . Because of this model’s rich behavior, it led to atures T and T , alternated with two constant-volume seg- à Carnot þ À a new field called finite-time thermodynamics,18 and is ments, which have non-isothermal heat processes. Pressure included in some textbooks.19 There is positive entropy pro- vs. volume and temperature vs. entropy graphs of the duction during each cycle. Stirling cycle are in Fig. 10.

The “first law efficiency” g does not account for limita- A. CCW Stirling cycle tions by the second law of thermodynamics. In contrast, a second law efficiency20  measures performance relative to 1. Reversible CCW Stirling cycle the maximum possible g using the same reservoirs and input The reversible Stirling cycle is well known and the CCW energy; i.e.,  g= 1 TL=TH 1. For example, at maxi- refrigerator was discussed recently by Mungan.22 The tempera- mum power the¼ Novikov-Curzon-Ahlbornð À Þ engine’s second ture range for both energy input Qin Q Qdc and output law efficiency is  1= 1 TL=TH . ¼ À þ Qout Q Qbc is the full interval T ; T . This is inconsis- ¼ ð þ Þ ¼ þ þ ð À þÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi tent with the specified narrow temperature range in the defini- tion of a refrigerator and the intended meaning of Eq. (6). From B. CCW Carnot cycle Fig. 10, it is clear that Qin Q Qdc exceeds the Carnot Carnot ¼ À þ 1. Reversible CCW Carnot cycle cycle’s input Q at T and WStirling, the shaded area adcba, À À is less than WCarnot, the area within the dashed rectangle. From Because Qin=T Qout=T ; the coefficient of perfor- Eq. (6) it follows that K K . The constant-volume þ ¼ À Stirling Carnot mance K can be expressed solely as a function of T ; T , segments “cancel” one another if the specific heat of the gas is À þ giving the following result, consistent with Eq. (7): volume dependent, with the reservoirs involved returning to Key Point 21. The reversible CCW Carnot cycle, a tradi- their initial states each cycle. If the intended goal for the device being modeled is cooling at temperature T , then using Qin tional refrigerator as described in Sec. IV, has coefficient À Q Qdc in Eq. (6) is misleading. A more relevant measure of performance KCarnot 1= T =T 1 . ¼ À þ ¼ ð þ À À Þ of efficiency is the task-specific coefficient of performance, 23 defined as Ktask Q =W < KStirling. In real-life Stirling À cycles, the movements of two pistons 2. Irreversible CCW Carnot cycle are coordinated to approximate the four Stirling cycle seg- ments, and with a regenerator, depicted in Fig. 11(a). As the If energy flows from the TL reservoir and to the TH reser- voir, it is necessary that the Carnot cycle’s low and high tem- constant volume gas winds its way through the regenerator peratures satisfy T < TL and T > TH, as in Fig. 9(b). The reservoir temperaturesÀ are assumedþ fixed and the working substance temperatures can be adjusted by design. In the double limit T TH; T TL, the reversible Carnot refrigerator is recovered.þ ! UnlikeÀ ! the heat engine, which has a maximum power condition, there is no such optimal operat- ing condition for the refrigerator. Key Point 22. Statements such as “heat flows uphill in a CCW cycle” are incorrect. The heat processes for the irre- versible cycle in Fig. 9(b) have normal energy flows from higher to lower temperatures, modeling real-life refrigera- tors; i.e., the hot coil is heated by compressive work, and becomes hotter than the kitchen, while the cold coil is cooled via expansion and is colder than the food Fig. 10. Two views of the Stirling cycle with alternating constant- compartment.21 temperature and constant-volume segments. (a) Pressure vs. volume. (b) Temperature vs. entropy. The dashed rectangle is for a comparison Carnot Despite irreversibility, the quasistatic Carnot cycle has cycle and the shaded lower rectangle is the energy exchanged between the Qin=T Qout=T and gas and reservoir at T . þ ¼ À À

350 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 350 VIII. CARNOT’S INEQUALITY FOR CW CYCLES A. Reversible CW cycles An interesting geometric demonstration of Carnot’s inequality for reversible CW cycles was presented by Tatiana Ehrenfest-Afanaseva in her book on thermodynamics.24 A variant, which incorporates adiabatic segments that separate the upper and lower temperature regions, is illustrated in Fig. 12. An arbitrary25 reversible cycle is represented by a closed path abcda (or adcba), circumscribed by a comparison Carnot cycle (dashed rectangle),26 chosen to have the arbi- trary cycle’s maximum and minimum temperatures. The thermal efficiency is defined as g 1 Qout=Qin, and Fig. 11. (a) Depiction of Stirling cycle apparatus with regenerator in which the ¼ À Qin uppera bT dS is the area under the upper path a b gas cools as the gas moves from the hot to cold side, and as the gas ¼ ! ! of the arbitrary cycle and the S axis. In Fig. 12(a), Qout is the moves oppositely. (b) Temperature profile of the gas in the regenerator, regard- Ð less of gas flow direction. Adapted from D. V. Schroeder, An Introduction to area of the hatched pattern, namely, the area under the lower Carnot Carnot Thermal Physics (Addison-Wesley-Longman, 1999), p. 133. path dc. Q and Q are the respective areas of the À þ lower shaded rectangle and rectangle efhg. Since Qin; Qout from higher to lower temperature, the regenerator retains a are non-negative by definition, it is clear visually that temperature gradient as in Fig. 11(b), generating path ba. Carnot Carnot Qin Q and Qout Q : (14) This process approximates using a sequence of successively  þ  À cooler reservoirs, ideally approaching reversibility. Using the latter inequalities, we have Similarly, heating along dc occurs when gas flows in the reverse direction. The CCW reversible Stirling cycle with Carnot Carnot g 1 Qout=Qin 1 Q =Q gCarnot: (15) regeneration uses only two reservoirs and has the same coef- ¼ À  À À þ ¼ ficient of performance as the reversible Carnot cycle. Key Point 24. Any reversible variable-temperature CW 2. Irreversible CCW Stirling cycle cycle represented in the temperature–entropy diagram in Fig. 12 satisfies g gCarnot, where the comparison Carnot For a quasistatic-irreversible CCW cycle, energy flows cycle circumscribes the arbitrary cycle. (from, to) the cold reservoir along (ad, ba), and (to, from) the hot reservoir along (cb, dc), so this is not a traditional refrig- erator. The net input from the cold reservoir per cycle is Qin B. Irreversible CW cycles Q Qba, so K Q Qba =W is more reflective of ¼ À À ¼ð À À Þ For any quasistatic CW cycle receiving Q and delivering the energy from the cold reservoir than Ktask. Irreversibility þ increases the environment’s entropy. Q from and to high and low temperature reservoirs, DScycle À0, and the total entropy change (of the universe) is the net entropy¼ change of the reservoirs B. CW Stirling cycle Q Q 1. Reversible CW Stirling cycle DS DS À þ 0: (16) tot ¼ cycle þ T À T  Analysis of the reversible clockwise Stirling cycle is sub- À þ stantially the same as for the counterclockwise case. Here Equation (16) reduces to Q =Q T =T , which implies Carnot Carnot À þ  À þ Qout Qcd Q Q and Qin Qab Q Q , Eq. (15). and it¼ followsþ fromÀ  Eq.À(5) and an area-based¼ þ argumentþ  þ using Fig. 10(b) that gStirling gCarnot. This cycle also can be used with a regenerator to approach reversible heat exchanges along the constant-volume seg- ments, in which case only the T and T reservoirs are involved, and the efficiency equalsþ the reversibleÀ Carnot effi- ciency, gCarnot. In analogy with the CCW case a, task-specific thermal efficiency gtask W=Q is sensible if the energy exchanges over the constant-volume þ segments cancel—or nearly do. In Stirling this case gtask gCarnot, the same result obtained with a regenerator. ¼ Fig. 12. (a) Temperature vs. entropy plot of an arbitrary CW or CCW revers- ible cycle (abcda or adcba) and a comparison Carnot cycle (dashed rectangle). 2. Irreversible CW Stirling cycle The vertical segments are quasistatic adiabatic (Q 0) paths that separate the ¼ Heating energy flows to the gas from the hot reservoir high and low temperature heat processes. A traditional heat engine or refrigera- tor cycle has Tb Ta and Td Tc much less than T ; T . For a CW cycle, along both ab and bc, and from the gas to the cold reservoir j À j j À j þ À Qab Qin, Qdc Qout;foraCCWcycle,Qab Qout, Qdc Qin.(b)Thesame along cd and da. Although not a traditional heat engine, this plot,¼ with a set¼ of narrow reversible Carnot cycle¼ rectangles¼ whose envelope cycle does supply external work. approximates the arbitrary cycle.

351 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 351 Key Point 25. Any irreversible CW quasistatic cycle, that of the reservoirs, DStot DScycle Qout=T Qin=T ¼ þ þ À À including those represented in Fig. 12 has g gCarnot, 0: The latter inequality together with Eq. (6) implies where the comparison Carnot cycle operates between the  arbitrary cycle’s maximum and minimum temperatures. K 1= Qout=Qin 1 1= T =T 1 : (20)  ð À Þ ð þ À À Þ

Key Point 27. For any 2-reservoir irreversible CCW cycle, IX. CARNOT-LIKE COEFFICIENT OF K KCarnot, an inequality that appears in some physics PERFORMANCE INEQUALITIES FOR CCW books .28 CYCLES

A. Reversible CCW cycles X. FINAL REMARKS I now present three different derivations leading to a gen- The examples above show how reversible and irreversible eralization of Eq. (8) for the reversible CCW Kelvin cycle. cycles that look the same on quasistatic pressure-volume or Each leads to Eq. (17), where the comparison Carnot cycle temperature-entropy graphs can result in very different temperatures are the arbitrary cycle’s extreme temperatures. behavior and entropy changes of the environment. The num- Derivation 1. From Fig. 12(a), Q =Q Qout=Qin, thus þ À  bered Key Points provide a capsule summary of major find- Eq. (6) leads to ings. Sections V–VII are illustrative of the differences between irreversible and reversible cycles. K KCarnot; generalizing Eq: 8 : (17)  ð Þ For irreversible cycles, the inequalities g gCarnot and K K are imposed by the second law of thermodynam- Derivation 2. The inequality (17) can also be obtained  Carnot using Fig. 12(b), where the ith approximating narrow Carnot ics. The g inequality holds also for non-Carnot reversible CW cycles. The K inequality for reversible CCW cycles, rectangle has energy input Qin;i and coefficient of perfor- K KCarnot, is not imposed by the second law because mance Ki. Note that each rectangular approximating Carnot  cycle has a larger coefficient of performance than the cir- reversible cycles generate zero entropy and always satisfy the second law. For the Stirling and other29 cycles with iso- cumscribing Carnot cycle; i.e., K K . This follows i  Carnot thermal segments, a task-specific coefficient of performance from Tout;i=Tin;i T =T , which implies Ki 1= Tout;i=  þ À ¼ ð Ktask and efficiency gtask can be useful efficiency measures. Tin;i 1 1= T =T 1 KCarnot. À Þ ð þ À À Þ¼ n We found that the reversible and irreversible CW Kelvin For the approximating Carnot rectangles, K j 1 Qin;j = n ¼ð ¼ Þ cycles have the same thermal efficiency gKelvin, which might ‘ 1 W‘ ,whereW‘ is the external work for the ‘th rectan- ð ¼ Þ P seem odd at first. However, the thermal efficiency g, which gle. Because Qin;j WjKj for the jth Carnot rectangle, P ¼ is a ratio of the work output to heat input, is unaffected by n Wj entropy-generating irreversibilities involving the reservoirs. K Kj KCarnot: (18) For the reversible case, the infinite set of reservoirs under- ¼ n  j 1 0 1 ¼ W‘ goes zero net entropy change, but for the irreversible case, X ‘ 1 B ¼ C the net entropy change of the hot and cold reservoirs is posi- @P A tive, as shown, e.g., in Eq. (10). The point is that the primary The first step represents K as a weighted average of the K j difference between the reversible and quasistatic irreversible for the approximating narrow Carnot cycles, and the secondf g cases lies in changes to the environment. step uses the previous result and the fact that, Kj KCarnot In summary, thermodynamic cycles are rich in insights n W = n W 1. Equation (18) confirms Eq. (8), ð j 1 jÞ ð ‘ 1 ‘Þ¼ and subtleties. Because there is much more to them than typ- which¼ was derived¼ for the Kelvin CCW cycle.27 P P ical textbook expositions indicate, they offer unique opportu- Derivation 3. The third proof of Eq. (17) comes from Eq. nities for teachers who would like to stimulate the mental (7),whichcanberearrangedtoreadK 1=g 1. For a ¼ À juices of their students. Carnot cycle, KCarnot 1=gCarnot 1. Using Carnot’s inequal- ity g g for the¼ reversible CWÀ cycle we obtain,  Carnot ACKNOWLEDGMENTS gCarnot g K KCarnot À 0 for reversible cycles; (19) The author is grateful to Carl Mungan, Don Lemons, and À ¼ gCarnotg  Jeffrey Prentis for their careful reading and excellent suggestions on drafts of this article. Additionally, the wise again agreeing with Eq. (8). The Carnot cycle reservoir tem- comments from two AJP reviewers led to significant peratures here are the maximum and minimum arbitrary improvements. The author has also benefitted from a spirited cycle temperatures used to obtain g g .  Carnot multi-year correspondence with Robert Dickerson and John Key Point 26. Any reversible CCW cycle has coefficient of Mottmann. performance K K . Its CW counterpart satisfies Carnot a) g g . For both cases, the comparison Carnot cycles Electronic mail: [email protected]; Present address: California State  Carnot Polytechnic University, 12705 SE River Road Portland, Oregon 97222. operate between the cycle’s maximum and minimum 1R. H. Dickerson and J. Mottman, “Not all CCW thermodynamic cycles are temperatures. refrigerators,” Am. J. Phys. 84, 413–418 (2016). 2R. Clausius, The Mechanical Theory of Heat (Macmillan and Co, London, 1879), Chap. X. B. Irreversible CCW arbitrary cycle 3See also, W. H. Cropper, “ and the road to entropy,” Am. J. Phys. 54, 1068–1074 (1986); E. Pellegrino, E. Ghibaudi, and L. Cerruti, For any quasistatic CCW cycle operating between T and “Clausius’ : A conceptual relic that sheds light on the second þ T , DScycle 0 for each cycle, and the net entropy change is law,” Entropy 17, 4500–4518 (2015). À ¼

352 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 352 4 d-Q, d-W connote inexact differentials, infinitesimals that sum to the non- 18B. Andresen, Finite-time Thermodynamics (Physics Lab II, Copenhagen, state functions Q and W. In contrast, dS d-Q=T is the exact differential 1983). of the entropy state function. ¼ 19D. V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 5J. D. Norton, “Thermodynamically reversible processes in statistical Reading, MA, 1999), p. 127. physics,” Am. J. Phys. 85, 135–145 (2017). 20M. H. Ross et al., Efficient Use of Energy, AIP Conf. Proc. No. 25 6Position-dependent temperatures and mole numbers are used in D. (American Institute of Physics, New York, 1975), pp. 27–35; T. V. Kondepudi and I. Prigogine, Modern Thermodynamics: From Heat Marcella, “ and the second law of thermodynamics: Engines to Dissipative Structures (John Wiley, New York, 2014). An introduction to second law analysis,” Am. J. Phys. 60,888–895 7See E. G. Mishchenko and P. F. Pshenichka, “Reversible temperature (1992). exchange upon thermal contact,” Am. J. Phys. 85, 23–29 (2017) for a 21President Harry Truman famously said “If you can’t stand the heat, get out totally different approach. of the kitchen.” For years I jokingly told students that President Truman 8W. Thomson (Lord Kelvin), “On the economy of the heating or cooling of really knew his thermodynamics. buildings by means of currents of air,” Proc. Roy. Philos. Soc. Glasgow 3, 22C. E. Mungan, “Coefficient of performance of Stirling refrigerators,” Eur. 269–269 (1852). J. Phys. 28, 055101 (2017). 9The subsequent invention of vapor compression brought 23The task-specific efficiency idea was suggested by a clever AJP referee. refrigeration and heat pump CCW cycles into prominence. 24T. Ehrenfest-Afanassjewa, Die Grundlagen der Thermodynamik (Brill, 10K. Seidman and T. R. Michalik, “The efficiency of reversible heat Leiden, 1956), p. 75; Believed to be out of print. See also M. C. Tobin, engines,” J. Chem. Ed. 68, 208–210 (1991). “Engine efficiencies and the second law of thermodynamics,” Am. J. Phys. 11 If x T =T < 1; DStotal 5Nk=2 x 1 ln x > 0 because x – 1, 37, 1115–1117 (1969). the straight¼ À lineþ tangent to ln¼ðx at x Þð1 liesÀ aboveÀ ð lnÞÞ x for x 1. 25This cycle is arbitrary within the class of cycles that transfer energy (from, 12H. S. Leff and G. L. Jones,ð “Irreversibility,Þ ¼ entropyð production,Þ 6¼ and ther- to) reservoirs along the upper path and (to, from) reservoirs along the mal efficiency,” Am. J. Phys. 43, 973–980 (1975). lower path for (clockwise, CCW) cycles, and are separated by adiabatic 13F. L. Curzon and B. Ahlborn, “Efficiency of a Carnot engine at maximum paths. power output,” Am. J. Phys. 43, 22–24 (1975). 26For reversible cycles, the maximum and minimum reservoir temperatures 14A. Bejan, “ advances on finite-time thermodynamics,” coincide with the arbitrary cycle’s maximum and minimum working sub- (Letter), Am. J. Phys 62, 11–12 (1994). stance temperatures. 15A. Bejan, “Models of power plants that generate minimum entropy while 27One can derive Eq. (15) for the CW case writing g as a weighted average, operating at maximum power,” Am. J. Phys 64, 1054–1059 (1996); An g f g g with f T DS = T DS , g 1 T =T , ¼ j j j  Carnot j ¼ in;j j ‘ in;‘ ‘ j ¼ À out;j in;j even older history has come to light: A. Vaudrey, F. Lanzetta, and M. H. B. and using Tin;j=Tout;j Tin=Tout. Feidt, “Reitlinger and the origins of the efficiency at maximum power for- 28See,P for example, p. 128 of Ref. 19. P mula for heat engines,” J. Non. Equilib. Thermodyn. 39, 199–203 (2014). 29The (https://en.wikipedia.org/wiki/Ericsson_cycle) has 16H. S. Leff, “Thermodynamics of combined-cycle plants,” alternating isothermal and constant-pressure segments and an argument Am. J. Phys. 80, 515–518 (2012). similar to that for the Stirling cycle can be made regarding compensation 17H. S. Leff, “Thermal efficiency at maximum work output: new results for of energy exchanges along the two non-isothermal segments. It can also be old heat engines,” Am. J. Phys. 55, 602–610 (1987). operated with a regenerator.

353 Am. J. Phys., Vol. 86, No. 5, May 2018 Harvey S. Leff 353