Three Factors Causing the Thermal Efficiency of a Heat Engine to Be Less Than Unity and Their Relevance to Daily Life

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Three factors causing the thermal efficiency of a heat engine to be less than unity and their relevance to daily life

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Three factors causing the thermal efﬁciency of a heat engine to be less than unity and their relevance to daily life

Jing Wu

School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China

E-mail: [email protected]

Received 1 August 2014, revised 25 August 2014 Accepted for publication 10 October 2014 Published 6 November 2014

Abstract The thermal efficiency of a heat engine is of primary interest in introductory thermodynamics courses. This study presents a simple method to quantitatively distinguish the factors that cause the thermal efficiency of a heat engine to be less than unity. These factors are the nonzero reference point, external irreversibilities and internal irreversibilities. Next, the difference between the present method and the existing second-law efficiency method for evaluating the influence of irreversibility is discussed. Then, an example of an actual heat- engine cycle is given to demonstrate the feasibility and effectiveness of the presented method. Finally, some remarks on the relevance of the three factors to our attitude toward life are presented.

Keywords: thermodynamics, thermal efﬁciency, irreversibility, availability, daily life

(Some ﬁgures may appear in colour only in the online journal)

1. Introduction

In thermodynamics, the ratio of the net work output of a heat engine to the total heat input to fl fi the working uid is called the thermal ef ciency, ηI, which is sometimes referred to as the first-law efficiency since it is defined on the basis of the first law (this is the reason why we use the subscript ‘I’ here). The thermal efficiency measures how efficiently a heat engine converts the heat that it receives to work. It is well known that the thermal efficiency of a heat engine is always less than unity; this is caused by various factors. However, the thermal efficiency itself makes no reference as to how each of the factors affects the thermal efficiency individually. For instance, if we are told

0143-0807/15/015008+12$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1 Eur. J. Phys. 36 (2015) 015008 JWu that a gas turbine has a thermal efficiency of 20 percent, we only know that the gas turbine converts 20 percent of the heat input to net work output. The relative magnitudes of various factors that may cause such a low thermal efficiency, such as the insufficiently large temperature difference between the hot and cold reservoirs or the irreversibilities involved in this gas-turbine cycle, are unknown. More importantly, by the thermal efficiency itself, we cannot evaluate separately the influences of internal irreversibilities and external irreversibilities, occurring within and outside the system boundary, respectively, on the thermal efficiency. In general, the thermal efficiency may be affected by the factors of the internal irreversibilities and external irreversibilities in quite different ways. Consider, as an example, an ideal Brayton cycle that operates between two given heat reservoirs. According to the definition of an ideal cycle [1, 2], there are no irreversibilities within the system boundary. Then, it can be concluded that the internal irreversibility is not the reason why the thermal efficiency of the ideal Brayton cycle is less than the corresponding Carnot efficiency. In fact, the reason is because this ideal Brayton cycle involves external irreversibilities due to heat transfer through a finite temperature difference between the working fluid and the two given reservoirs. Then, the question that arises naturally is how to quantitatively distinguish different factors that cause the thermal efficiency of a heat engine to be less than unity, especially the factors of external and internal irreversibilities. This study focuses on this question. The thermal efficiency of a heat engine is expressed as a product of the Carnot efficiency, the external irreversibility factor and the internal irreversibility factor, by which the three factors that affect the thermal efficiency can be clearly distinguished, and the influences of the external and internal irreversibilities on the thermal efficiency can be further separated from each other. Next, an example of an actual heat engine is given to demonstrate the feasibility and effectiveness of the presented method to distinguish different factors. In addition, it is found that many students view thermodynamics only as an engineering subject. As a result, many basic principles or concepts in thermodynamics, such as the second law of thermodynamics or the concept of thermal efficiency, only have ‘engineering’ meanings in their minds. This is unfortunate because it causes some students to grasp only one side of thermodynamics and miss the complete picture. In fact, many thermodynamic principles or concepts can serve as excellent vehicles to help us understand some aspects of daily life since they are ‘universal’ rather than specific only to ‘technical’ fields. The extension of the thermodynamic principles or concepts to nontechnical fields can be found in several papers or books [1, 3–6]. However, to the author’s best knowledge, the relevance of the factors that affect the thermal efficiency of a heat engine to our attitude toward life has not been reported previously. Thus, in section 4 of this study, we attempt to demonstrate this relevance. Hopefully, it will contribute to a better understanding and appreciation of the principles and concepts in thermodynamics and will encourage us to use the abstract concepts in thermodynamics more often in technical and even nontechnical areas. This study is firstly intended for undergraduate or graduate students and teachers with some knowledge of thermodynamics; however, we hope that even readers with nontechnical backgrounds will appreciate the life-related part of this study to stimulate their interest in having a basic or even a detailed understanding of thermodynamics.

2. Three factors combine to determine thermal efﬁciency

To separate the factors that affect the thermal efﬁciency of a heat engine, we express the ﬁ thermal ef ciency, ηI,as

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Figure 1. The Stirling and Ericsson cycles have the same thermal efﬁciency as the Carnot cycle between the same temperature limits (Tmin and Tmax)[1].

ηI=⋅ηη C ext,ir ⋅ η int,ir,(1) ﬁ where ηC is the well-known Carnot ef ciency, and ηext,ir and ηint,ir are, respectively, named the external irreversibility factor and internal irreversibility factor in this study. The above expression is valid for frequently encountered heat-engine cycles operating between two reservoirs in introductory thermodynamic textbooks, whatever their details of operation. Notice that none of ηC, ηext,ir and ηint,ir can exceed 100 percent. Next, we discuss them in detail one by one.

η 2.1. The Carnot efficiency ( C) ﬁ ﬁ We know that the Carnot ef ciency, ηC, is the highest thermal ef ciency that a heat engine operating between the two reservoirs at temperatures Tmin and Tmax can have, which is given by

Tmin ηC =−1. (2) Tmax

In a heat-engine cycle, Tmin can be viewed as the reference-point temperature, and its value is limited by cooling mediums such as rivers, lakes or the atmosphere. Since Tmin ≠ 0, ηC is always less than 100 percent. Hence, it is the nonzero reference point that makes it impossible for even a ‘perfect’ engine to have a 100 percent thermal efficiency. There are two other cycles that have the Carnot efficiency: the Stirling cycle and Ericsson cycle, as shown in figure 1. They differ from the Carnot cycle in that the two isentropic processes are replaced by two constant-volume regeneration processes in the Stirling cycle and by two constant-pressure regeneration processes in the Ericsson cycle. However, they are both totally reversible cycles, as is the Carnot cycle, because any heat transfer between the working fluid and the regenerator is through a differential temperature difference. Thus, it can be concluded that only in the circumstance of totally reversible circum- stances, namely no internal and external irreversibilities during the heat-engine cycle, will a fi Tmin heat engine have the Carnot ef ciency, ηC =−1 . Tmax

η 2.2. The external irreversibility factor ( ext;ir) In thermodynamics, the actual cycles are usually somewhat idealized to make an analytical study of a cycle feasible. The internal reversible cycle, namely the ideal cycle [1, 2], is an idealized one. All of the internal irreversibilities that an actual cycle may involve, such as nonquasi-equilibrium changes, friction and other dissipative effects, are neglected. Figure 2 shows the T–s diagrams of the frequently encountered ideal cycles in introductory thermodynamic textbooks, including the Otto cycle for the gasoline engines, the Diesel cycle for the

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Figure 2. Some of the frequently encountered ideal cycles in introductory thermodynamic textbooks between the same temperature limits Tmax and Tmin. diesel engines, the Brayton cycle for gas-turbine engines and the Rankine cycle for steam- turbine engines. The ideal cycles are internally reversible, but unlike the Carnot cycle, they are not necessarily totally reversible since they may involve irreversibilities that are external to the system. For instance, the ideal Otto, Diesel, Brayton or Rankine cycles operating between a high-temperature reservoir at Tmax and a low-temperature reservoir at Tmin are externally irreversible due to the heat transfer through a finite temperature difference during the heat- addition process 2–3 or heat-rejection process 4–1, as shown in figure 2. As a result, it is known that the thermal efficiency of an ideal cycle is, in general, less than that of a totally reversible cycle (i.e. the Carnot, Stirling or Ericsson cycles) operating between the same temperature limits (Tmin and Tmax). The more closely the ideal cycle approximates the totally reversible cycle, the higher thermal efficiency the ideal cycle has. Thus, it would be desirable to have a parameter that expresses quantitatively the degree of approximation of an ideal cycle to the corresponding totally reversible cycle between the same temperature limits. This parameter is named the external irreversibility factor, designated ηext,ir, and is expressed as

ηideal ηext,ir = ,(3) ηC fi where ηideal is the thermal ef ciency of an ideal cycle, and ηC is that of the corresponding Tmin totally reversible cycle, namely ηC =−1 . Tmax The external irreversibility factor, ηext,ir, enables us to compare the performance of different ideal cycles, which are designed to operate between the same temperature limits, indicating how the path design of an ideal cycle affects its thermal efficiency: For a ‘ther- ’ modynamically perfect design, such as the totally reversible Carnot cycles, ηext,ir = 1, whereas for other designs, as long as the path the working fluid follows is not totally fi reversible, their thermal ef ciency drops from ηC to ηC⋅ η ext,ir due to the external irreversibilities. The farther away ηext,ir is from unity, the more the ideal cycle deviates from the totally reversible one. It is worth noting that in this study, we assume that there is only one high-temperature reservoir and one low-temperature reservoir, as mentioned after equation (1). Under this condition, the ideal cycles, as shown in figure 2, are externally irreversible. When the number of heat reservoirs that the heat engine contacts during one cycle is larger than two, however, these ideal cycles can be operated in a totally reversible way because any heat transfer between the working fluid and the heat reservoir can occur through a differential temperature difference by introducing an infinite number of heat reservoirs. Under the condition of numerous reservoirs, the form ηI=⋅ηη C ext,ir ⋅ η int,ir can still be retained, but the external irreversibility factor, ηext,ir, is not appropriate here since the ideal cycle is no longer externally

4 Eur. J. Phys. 36 (2015) 015008 JWu irreversible. The expression ηI=⋅ηη C ideal path ⋅ η int,ir is probably more appropriate under this condition, for which ηideal path implies the effect of the path design of an ideal cycle, which is also determined by equation (3); in other words, ηideal path= ηη ideal C. The farther away ηideal path is from unity, the more the ideal cycle deviates from the corresponding Carnot cycle with the same temperature limits.

η 2.3. The internal irreversibility factor ( int;ir) Like the external irreversibilities, the internal irreversiblities that occur within the system boundary can also decrease the thermal efficiency of an actual cycle. As discussed in section 2.2, the effect of external irreversibility on the thermal efficiency can be quantitatively accounted for by ηext,ir. Likewise, we can introduce the internal irreversibility factor, designated ηint,ir, to account for the extent to which an actual heat-engine cycle approaches the corresponding ideal cycle, defined as

ηI ηint,ir = ,(4)a ηideal ﬁ where ηI is the thermal ef ciency of an actual heat-engine cycle. Substituting the relation ηideal=⋅ηη C ext,ir from equation (3) into equation (4a) yields

ηI ηint,ir = ,(4)b ηηC⋅ ext,ir which is equivalent to equation (1), showing that the third term in the product on the right- hand side of equation (1) is just the internal irreversibility factor, ηint,ir. Clearly, ηint,ir is unity for an ideal cycle that involves no internal irreversibilities. The farther away ηint,ir is from unity, the more the actual cycle deviates from the corresponding ideal one. In summary, the three factors that cause the thermal efficiency to be less than unity are (figure 3): 1. The nonzero reference point (Tmin) This factor is accounted for by the Carnot efficiency, which can be understood as the ‘reference-point efficiency’. It is naturally inevitable because of the unreachable absolute zero temperature. For many heat engines, the reference point is restricted by the environment since rivers, lakes or the atmosphere are usually taken as the cooling medium, and Tmin equals the environmental temperature, T0. Note that this factor is induced by unavailability rather than irreversibility. 2. The external irreversibilities

This factor is quantitatively accounted for by the external irreversibility factor, ηext,ir, which has a value that essentially depends on the choice of the ideal path. In contrast to the first factor, which is naturally inevitable, this factor is somewhat ‘controllable’ since it is possible, at least theoretically, for the designers to make the ideal path approach the totally reversible one (i.e. the Carnot, Stirling or Ericsson cycle) despite the economical limitations and impracticalities. For instance, in gas-turbine engines, the simple Brayton cycle has a very low thermal efficiency due to its low external irreversibility factor. However, if the multistage compression with intercooling and multistage expansion with reheating are used in con- junction with regeneration, the thermal efficiency of the Brayton cycle increases as a result of the increased external irreversibility factor. As the number of compression and expansion stages increases, the external irreversibility factor approaches unity. Consequently, the gas-

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Figure 3. Graphical representation of the three factors due to which heat supplied to the working ﬂuid cannot be entirely converted to work by a heat engine operating between two given reservoirs.

Figure 4. Ericsson cycle as a limit of the ideal Brayton cycle operating using multistage compression with intercooling, multistage expansion with reheating and regeneration [2]. turbine cycle with intercooling, reheating and regeneration approaches the Ericsson cycle, as illustrated in ﬁgure 4, and it possesses the same thermal efﬁciency as the Carnot cycle. 3. The internal irreversibilities

This factor is quantitatively gauged by the internal irreversibility factor, ηint,ir. Like the second factor, it is also somewhat ‘controllable,’ although it is impossible to completely

6 Eur. J. Phys. 36 (2015) 015008 JWu eliminate all of the internal irreversibilities in practice. Note that this factor, together with the second one, is induced by the imperfect use of the available part of heat. We know that although a totally reversible cycle between two heat reservoirs has the maximum thermal efficiency, its power output approaches zero, which is obviously mean- ingless for engineering applications. This is because all of the heat transfer processes involved in the cycle need to be through a differential temperature difference to ensure total reversi- bility, which results in an infinitely long operation time and zero power (power = work/time). In order to obtain a certain amount of power, the irreversibility of finite-time heat transfer processes is proposed. This method of modeling and optimizing a real thermodynamic cycle was referred to as finite time thermodynamics in physics and engineering literature; the main goal of this method is to obtain the performance bounds and optimal criteria of selecting thermodynamic parameters for heat engines, heat pumps and refrigerators [7, 8]. The irreversibilities of a heat engine considered in finite time thermodynamics can also be reflected by the above-mentioned external and internal irreversibility factors. For instance, for the endoreversible C-A model [9], we have ηext,ir < 1 and ηint,ir = 1 since the irreversibility of this model is due only to the heat transfer through a finite temperature difference between the working fluid and the reservoirs. However, for Novikov’s model, in addition to the external irreversibility due to the heat transfer, the internal irreversibility in the expansion process is also considered [10]. Hence, we have ηext,ir < 1 and ηint,ir < 1.

2.4. A comparison with the existing second-law efficiency fi In thermodynamics, the second-law ef ciency, ηII, is a measure of the actual performance relative to the best possible performance under the same conditions [11, 12]. For a heat- engine cycle, it is defined as the ratio of the actual thermal efficiency to the maximum possible (reversible) thermal efficiency [12]. As discussed in section 2.1, the maximum fi fi possible thermal ef ciency is the Carnot ef ciency, ηC, for all of the heat engines with two given reservoirs at Tmin and Tmax. Therefore, the second-law efficiency of an actual heat engine executed between these two reservoirs is

ηI ηII = .(5)a ηC Comparing it with equation (1) yields

ηII=⋅ηη int,ir ext,ir.(5)b

ﬁ This equation shows that as a product of ηint,ir and ηext,ir, the second-law ef ciency, ηII, includes both the external and internal irreversibility factors. However, their individual effects ﬁ on the thermal ef ciency cannot be separated by ηII itself. In fact, this is the reason that led us introduce the external irreversibility factor and internal irreversibility factor separately in this study.

3. Case study and discussion

Consider an actual gas-turbine cycle operating between two reservoirs at temperatures T3 and T1, as illustrated in figure 5. This cycle has a pressure ratio of π = 14. The gas temperature is 300 K at the compressor inlet and 1600 K at the turbine inlet. The lowest pressure of this cycle is 100 kPa. Assuming a compressor efficiency of 0.85, a turbine efficiency of 0.88, a constant specific heat of cp =⋅1.004 kJ/(kg K) and a constant specific heat ratio k = 1.4, one

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Figure 5. A T–s diagram of an actual gas-turbine cycle.

can determine the relative magnitudes of the three factors ηC, ηext,ir and ηint,ir discussed in section 2. ﬁ 1. The Carnot ef ciency ηC The highest and lowest temperatures of the actual cycle are T3 = 1600 K and T1 = 300 K, respectively. Thus, the thermal efﬁciency of a Carnot cycle operating between T1 and T3 is

Tmin T1 300 K ηC =−111 =− =− =0.8125. Tmax T3 1600 K

2. The external irreversibility factor ηext,ir The corresponding internally reversible cycle of the actual cycle is the Brayton one (1–2s–3–4s–1), which has a thermal efﬁciency that can be determined by

1 η =−1 .(6) ideal Brayton k−1 π k

Substituting the known numerical values (π = 14 and k = 1.4), we have ηideal Brayton = 0.53. Thus, the external irreversibility factor of this actual gas-turbine cycle is ηideal Brayton 0.53 ηext,ir ===0.65. ηC 0.8125 This value reflects the degree of approximation of the ideal Brayton cycle to the corresponding Carnot cycle between the same temperature limits (T1 and T3). Obviously, fi for xed values of T1, T3 and k, as the pressure ratio, π, is increased, ηext,ir increases and finally approaches unity; in other words, the ideal Brayton cycle approaches the Carnot cycle, as shown in figure 6.

3. The internal irreversibility factor ηint,ir To obtain the internal irreversibility factor ηint,ir, we need to determine the thermal efficiency of the actual gas-turbine cycle first. The air temperatures at the compressor and turbine exits, T2 and T4, are determined from isentropic relations and the definitions of the compressor and turbine efficiencies, as follows: For isentropic compression process 1–2s,

p21==×π p 14 100 kPa = 1400 kPa

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Figure 6. As the pressure ratio, π, increases, the ideal Brayton cycle approaches the Carnot cycle between the same temperature limits T1 and T3.

γ−1 ⎛ p ⎞ γ ⎜ 2 ⎟ TT21s = ⎜ ⎟ ⎝ p1 ⎠ γ−1 = T1π γ 1.4− 1 =×300 K 14 1.4 = 637.63 K. ﬁ ﬁ Using the de nition of compressor ef ciency, ηC,s, gives

TT21s − TT21=+ ηC,s 637.63 K− 300 K =+300 K 0.85 = 697.2 K. – For isentropic compression process 3 4s, p4s = 100 kPa and

γ−1 ⎛ p ⎞ γ ⎜ 4s ⎟ TT43s = ⎜ ⎟ ⎝ p3 ⎠ γ−1 ⎛ ⎞ γ ⎜⎟1 = T3 ⎝ π ⎠ 1.4− 1 ⎛ 1 ⎞ 1.4 =×1600 K ⎜⎟ ⎝ 14 ⎠ = 752.76 K.

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ﬁ ﬁ Using the de nition of turbine ef ciency, ηT, we have

TT43=−ηT () TT34 −s =−1600 K 0.88 ×−(1600 K 752.76 K) = 854.43 K. Thus, the thermal efﬁciency is

wnet ηI = qin q =−1 out qin hh− =−1 41 hh32−

cTp ()41− T =−1 cTp ()32− T = 0.386. Then, from equation (4b),

ηI ηint,ir = ηηC⋅ ext,ir 0.386 = 0.8125× 0.65 = 0.73, which reflects the effect of internal irreversibility of this actual gas-turbine cycle on the thermal efficiency. The calculation above shows that due to the nonzero reference point, only 81.25 percent of the heat absorbed by the working fluid is available, and about 65 percent of this amount of available energy may be converted into work output since the operating path the designer chose is a Brayton cycle rather than a Carnot one. Moreover, the internal irreversibilities during the compression and expansion processes results in a further decrease of the thermal efficiency; in other words, only 38.6 percent of the heat input is finally converted to work output, as shown in figure 7.

4. An inspiration for attitude to life

The three different factors that affect the thermal efﬁciency discussed above might shed some light on our attitude to life. There is no doubt that life is full of loss, upsets, setbacks, disappointments, difﬁculties, trials, tribulations and all that not-so-good stuff. It is just as the saying goes that ‘life is not a bed of roses’. The corresponding version in China is that ‘eight or nine out of ten things in our life cannot always be satisfactory’. So, how do we face and deal with the not-so-good things in a more unhurried, philosophical and positive manner? First, we have to accept the fact that in many situations, the not-so-good things in our daily life are not in our control. They are restricted by our surroundings, such as the society we live in, the family we grew up with or the personal talents we possess. Despite all of our efforts, wisdom and resourcefulness, we are still not able to obtain perfect results, as we

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Figure 7. As a result of the nonzero reference point, external irreversibilities and internal irreversibilities, the part of the heat input that can be converted to net work output decreases gradually; finally, only 38.6 percent of the heat input is converted to work output. would expect. This is just like the Carnot efficiency discussed in section 2.1: even for a ‘perfect’ engine, in the absence of any irreversibility, its thermal efficiency can never reach 100 percent due to the restriction of its surroundings—the nonzero reference point. Thus, we should accept the imperfection of life no matter how much pain and suffering we may experience during this process, just as Jack said to Rose in the movie Titanic (Cameron, 1997), ‘You learn to take life as it comes at you’. Second, under the restriction of our surroundings, what we can do is be prepared to use all of our abilities to deal with the things in our control: we can choose a path among several alternatives that approaches the best one (which can be viewed as the line of thinking) and, at the same time, avoid mistakes as much as possible (which can be viewed as the details of execution). This is just like the two approaches to increasing the thermal efficiency of a heat engine, as discussed in sections 2.2 and 2.3, respectively: since some of the heat is not available, what we can do is utilize the remaining available part to the best of our ability by choosing a cycle path that approaches the Carnot, Stirling or Ericsson cycle to the extent possible and decreasing the internally irreversible loss. As Ben Stein (the American actor, lawyer and writer) said, ‘The successful people of this world take life as it comes. They just go out and deal with the world as it is’.

5. Concluding remarks

The thermal efﬁciency of a heat-engine cycle operating between two reservoirs can be ﬁ expressed as ηI=⋅ηη C ext,ir ⋅ η int,ir, where the Carnot ef ciency, ηC, represents the degree of availability of heat input that can be reached only by totally reversible cycles such as the

Carnot, Stirling and Ericsson cycles; the external irreversibility factor, ηext,ir, measures the degree of approximation of an ideal cycle to the corresponding totally reversible cycle between the same temperature limits; in addition, the internal irreversibility factor, ηint,ir, gauges the extent to which an actual heat-engine cycle approaches the corresponding ideal one. The nonzero reference point, the external irreversibilities (i.e. the choice of the ideal path) and the internal irreversibilities, quantitatively accounted for by ηC, ηext,ir and ηint,ir, respectively, are the three factors that cause the thermal efﬁciency to be less than unity. Unlike the

11 Eur. J. Phys. 36 (2015) 015008 JWu factor of the nonzero reference point, which is naturally inevitable and uncontrollable, the two other factors are, at least theoretically, controllable despite the economical limitations and impracticalities. Furthermore, a comparison analysis indicates that the existing second-law efficiency in thermodynamics includes both the external and internal irreversibility factors, but their individual effects on the thermal efficiency cannot be quantitatively separated by the second-law efficiency. Finally, we attempt to show the relevance of the factors that affect the thermal efficiency of a heat engine to our attitude toward life. On the one hand, we should accept the imperfection of life, which is mostly restricted by our surroundings and is not in our control, just like the nonzero reference point, which is naturally inevitable. On the other hand, under the restriction of our surroundings, what we can do is be prepared to use all of our abilities to deal with the things in our control, including choosing a path that approaches the best one and avoiding mistakes as much as possible, just like the two approaches to increasing the thermal efficiency of a heat engine: choosing a cycle path that approaches a totally reversible one and decreasing the internally irreversible loss.

Acknowledgments

The author would like to thank Prof. Z Y Guo for his helpful discussions. This work is supported by the National Natural Science Foundation of China (51206079) and the Fun- damental Research Funds for the Central Universities (2014TS116).

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