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Performance of irreversible heat engines at minimum entropy generation
Applied Mathematical Modelling 37 (2013) 9810–9817
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Short communication
Performance of irreversible heat engines at minimum entropy generation Y. Haseli ⇑ Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, Eindhoven 5600 MB, The Netherlands
a r t i c l e
i n f o
Article history: Received 24 August 2012 Received in revised form 21 May 2013 Accepted 23 May 2013 Available online 5 June 2013 Keywords: Heat engine Entropy generation Efficiency Work output Finite time Thermodynamics
a b s t r a c t A thermodynamic analysis is presented by means of mathematical formulation to examine the performance of the most common types of heat engines including Otto, Diesel, and Brayton cycles, at the regime of minimum entropy generation. All engines are subject to internal and external irreversibilities. It is shown that minimum entropy production criterion neither correlates with maximum thermal efficiency design nor with maximum work output criterion. The results demonstrate that the production of entropy is not necessarily equivalent to the energy losses taking place in real devices. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The topic of heat engines is an old subject of thermodynamics. The optimization of energy conversion systems; i.e. heat engines, has received remarkable attention by numerous scholars, in particular, since 1970s. Various optimization objectives have been employed to investigate the performance of well-known standard thermodynamic cycles including Brayton, Otto, Diesel, Atkinson, Miller, and Dual cycles. The most familiar optimization criteria examined in past works are maximum efficiency, maximum work (power) output, minimum entropy generation, and maximum ecological function (defined as power minus entropy generation rate times environment temperature). The methodology employed in past works for optimization of heat engines is based on Finite Time Thermodynamics (FTT), which suggests that to model a realistic operation of engines, it is necessary to account for finite time operation of the components of a given power cycle. This, unlike fully reversible cycles; e.g. Carnot, which consists of a sequence of infinitesimal processes, requires the various components of a power cycle to run in a finite time in order to produce a desired amount of power (work). The most practical optimization objectives in real life applications of power cycles are the thermal efficiency and the power output of a given engine. The design point corresponding to maximum efficiency is not necessarily identical to that of maximum power output. For instance, the maximum thermal efficiency and maximum work output for a regenerative Brayton-type heat engine are coincident only at a regenerative heat exchanger efficiency of 50% [1–3]. At the regime of maximum efficiency, we aim to extract as much work as possible per unit mass of fuel. On the other hand, at the regime of maximum work output, we aim to extract maximum possible power from engine without regard to how much fuel is spent. Whether the optimization should be performed according to maximum power or maximum efficiency depends on the problem constraints; the relative prices of power and fuel [4]. ⇑ Corresponding author. Present address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139, USA. E-mail addresses: [email protected], [email protected]. 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.05.010
Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
9811
The question of interest is when a heat engine is optimized based on a criterion other than maximum efficiency or maximum work output, what would be the efficiency and the net work production of the engine? In particular, if one designs an engine based on minimization of entropy production criterion, what would be the thermal efficiency and power output at the corresponding regime? To answer this question is the main objective of the present paper. Three well-known standard thermodynamic cycles including Brayton, Otto, and Diesel cycles which experiences internal and external irreversibilities are investigated to find whether minimization of entropy production associated with the operation of these engines may lead to conclusive design guidelines. The above question has somewhat been the interest of few authors. Indeed, in 1975, Leff and Jones [5] discussed by means of analytical argument that an increase in the thermal efficiency of an irreversible heat engine would not necessarily result in a decrease in its entropy production. Salamon et al. [6,7] argued in early 1980s that maximum work and minimum entropy production are two different designs in the optimization of heat engines. In 1996, Bejan [8] raised objection to the statement of Salamon et al. [6,7] arguing that according to Gouy–Stodola theorem maximum power output is equivalent to minimum entropy production rate. In his discussions, he pointed out the simplest and oldest power plants models of Curzon and Ahlborn [9], Chambadal [10] and Novikov [11], whose efficiency at maximum power operation is represented by gCA = 1 (TL/TH)½. Bejan argued that this efficiency could also be achieved by minimizing the entropy generation rate associated with the operation of the above models of power plants. Five years after Bejan’ criticism, Salamon et al. [4] argued through a comprehensive discussion that the justification of Bejan [8] about the equivalence of maximum work and minimum entropy production in heat engines is limited to only certain design conditions. An important observation made by Salamon et al. [4] was the method of evaluation of the total entropy production rate associated with the operation of a power producing compartment. In a recent book on power plants optimization [12], the power plant models of Curzon and Ahlborn [9], Novikov [11], modified Novikov [12], and Carnot vapor cycle is examined in detail. It is shown that the regime of minimum entropy production is different from that of maximum work output for the above first three models. For the case of a Carnot vapor cycle, which exchanges heat with a hot stream and a cold stream fluids through two heat exchangers at the hot-end and the codend sides of the engine, if the heat input is fixed and the energy contents of both hot and cold exhaust streams are not utilized in a secondary application, the operational regimes of minimum entropy production, maximum work output and maximum efficiency are identical. As mentioned previously, the central focus of the present paper is to examine the performance of three common power cycles; i.e. Brayton, Otto and Diesel, at the condition of minimum entropy production. In 1987, Leff [13] discovered that assuming constant specific heats, the efficiency of the standard air-cycles including Otto, Diesel, Joule-Brayton and Atkinson, at maximum work output could be represented exactly or approximately with Curzon–Ahlborn efficiency [9]. Leff had further assumed that the maximum and minimum temperatures of the above mentioned cycles were TH and TL, respectively. This paper aims to examine the performance of these heat engines at minimum entropy generation. The engines examined by Leff [13] and Cruzon and Ahlbon [9] are typical examples of endo-reversible cycles since the compression and expansion processes are assumed to be isentropic. In practical applications, however, it is almost unavoidable to neglect internal irreversibilities. A more realistic analysis would therefore require one to account for the irreversibilities of the compression and expansion processes together with the irreversible heat transfer processes between the engine and the high and low temperature reservoirs due to the finite temperature differences. Thus, we examine the performance of irreversible Otto, Diesel and Brayton cycles whose T–S diagram is depicted in Fig. 1. These engines undergo irreversible compression and expansion processes through lines 1–2 and 3–4, respectively. The isentropic compression and expansion processes are shown by dotted lines 1–2s and 3–4s, respectively. Line 2–3 represents a heat transfer process from the high temperature reservoir (TH) to the engine at constant volume/pressure/pressure in the Otto/Diesel/Brayton cycle. Line 4–1 shows the rejection of heat to the surrounding maintained at temperature TL at constant volume/volume/pressure in the Otto/Diesel/Brayton cycle. The isentropic efficiencies of the compression and the expansion processes are represented as gcom ¼ ðT 2s T 1 Þ=ðT 2 T 1 Þ and gexp ¼ ðT 3 T 4 Þ=ðT 3 T 4s Þ, respectively. Similar to the work of Leff [13], the analysis will be based on certain simplifying assumptions including, constant specific heat throughout the cycle, negligible mass of fuel compared to that of working gas, and an ideal gas behavior of the working substance (air). The maximum and minimum temperatures of the engine; i.e. T3 and T1, are the problem constraints due to the metallurgical limitation of the material of engines, and ambient conditions, since air is supplied from the atmosphere to the engines. 2. Mathematical formulation 2.1. Otto cycle Let us first consider the Otto cycle. As a primary step for evaluating the performance of the engine in terms of efficiency and work output, we need to determine temperatures T2 and T4 as functions of other operating parameters. Noting that ðT 3 =T 4s Þ ¼ ðT 2s =T 1 Þ ¼ r c1 , we get
T2 ¼ T1 þ
T 2s T 1
gcom
r c1 1 ; ¼ T1 1 þ
gcom
ð1Þ
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
TH
T
3
QH
4 4s 2 2s
QL 1
TL s Fig. 1. T–S diagram of irreversible Otto, Diesel and Brayton cycles.
h i T 4 ¼ T 3 gexp ðT 3 T 4s Þ ¼ T 3 1 gexp 1 r1c ;
ð2Þ
where r is the compression ratio defined as r = V1/V2 and c = cP/cV. The heat input and the heat output per unit mass of the air are obtained from
r c1 1 ; Q H ¼ c V ðT 3 T 2 Þ ¼ c V T 1 r T 1
ð3Þ
h i Q L ¼ cV ðT 4 T 1 Þ ¼ cV T 1 rT 1 gexp r T 1 r1c ;
ð4Þ
gcom
where rT = T3/T1. The work output and the thermal efficiency are therefore determined as follows.
r c1 1 ; ðW net ÞOtto ¼ Q H Q L ¼ cV T 1 rT gexp 1 r 1c
gcom
gOtto ¼
ðW net ÞOtto r T gexp gcom 1 r1c r c1 1 ¼ : QH ðrT 1Þgcom ðr c1 1Þ
ð5Þ
ð6Þ
Next, we evaluate the total entropy production associated with the operation of the engine. Entropy is generated due to four irreversible processes; i.e. the compression process (path 1–2), the heat transfer to the engine (path 2–3), the expansion process (path 3–4), and the heat rejection from the engine (path 4–1) to the ambient. Hence,
Sgen ¼
QL QH : TL TH
ð7Þ
Substituting Eqs. (3) and (4) into Eq. (7) and arranging yields
rc1 1 Sgen Otto ¼ cV ðr T 1Þð1 pÞ gexp rT 1 r 1c þ p ;
gcom
ð8Þ
where p = TL/TH. Notice that in the present analysis, TL = T1. To minimize the total entropy generation, we apply @Sgen =@r ¼ 0 whose solution gives
ðr ÞSgen; min ¼
r
T
p
gexp gcom
2c12
:
ð9Þ
We can also find an optimum compression ratio which would result in a maximum work output by applying ð@W net =@r Þ ¼ 0 to Eq. (5). This leads to
2c12 ðr ÞW max ¼ r T gexp gcom :
ð10Þ
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
This results shows that the work output is zero at a certain compression ratio that is greater than ðr ÞW max ; which can be obtained by solving Eq. (5) with ðW net ÞOtto ¼ 0. Hence, 1
c1 ðr ÞW net ¼0 ¼ rT gexp gcom ¼ ðr Þ2W max :
ð11Þ
From Eqs. (9) and (10) it can be inferred that
ðr ÞSgen; min ¼ ðr ÞW max
2c12 1
p
ð12Þ
:
Equation (12) reveals that the operational regimes at maximum work output and minimum entropy generation are different. A further subtle observation can be made by comparing Eqs. (9) and (11): it can be readily shown that ðr ÞSgen; min > ðr ÞW net ¼0 . In other words, the work output of the engine at the regime of minimum entropy production would be negative! The maximum compression ratio that the engine may approach is ðrÞW net ¼0 . This argument leads us to conclude that it is desirable for the Otto cycle to operate at minimum entropy generation, and the efficiency at maximum work would be more desirable. 2.2. Diesel cycle For the case of the Diesel cycle, we undertake a similar procedure. We may still use Eq. (1) for determination of T2. However, T4 is obtained as follows.
(
"
T 4 ¼ T 3 1 gexp 1
r rcut
1c #) ;
ð13Þ
where 1 < rcut < r, and rcut denotes the cut-off ratio defined as rcut = V3/V2 > 1. It is related to the temperature ratio rT as
rT ¼ r c1 r cut :
ð14Þ
Consequently, we get the following expressions for the work output, the efficiency and the total entropy generation of the Diesel cycle.
(
"
ðW net ÞDiesel ¼ cV T 1 ðc 1Þðr T 1Þ þ gexp r T 1
r
1c #
r cut
r c1 1
gcom
)
c ;
ð15Þ
1c c1 r r g 1 c ðc 1Þðr T 1Þ þ gexp r T 1 rcut com
¼ ; r c1 1 c rT 1 g
gDiesel
ð16Þ
com
Sgen
Diesel
(
"
¼ cV ðrT 1Þð1 cpÞ gexp r T 1
r
r cut
1c # þ
r c1 1
gcom
)
cp :
ð17Þ
Maximization of the work output and minimization of the entropy generation result in the following optimum compression ratios.
21 ðr ÞW max ¼ r cT gexp gcom c 1 ;
ðr ÞSgen; min ¼
c rT
p
gexp gcom
ð18Þ
21
c 1
¼ ðr ÞW max
21 1 c 1
p
ð19Þ
:
As rcut > 1, a combination of Eqs. (14) and (19) leads to the following inequality.
rT >
gexp gcom ; p
ð20Þ
which implies that for any rT less than gexpgcom/p, minimization of entropy generation would lead to an rcut less than 1, which by definition is incorrect. For instance, assume gexp = 0.9, gcom = 0.85, and p = 0.18. For r T < 4:25ð¼ 0:9 0:85=0:18Þ, a design based on minimum entropy production criterion would require rcut < 1! 2.3. Brayton cycle c1
For the Brayton cycle, the temperatures T2 and T4 are obtained using the relationship ðT 3 =T 4s Þ ¼ ðT 2s =T 1 Þ ¼ r pc as follows.
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
0
1 c1 c r 1 p A; T 2 ¼ T 1 @1 þ
ð21Þ
1c ; T 4 ¼ T 3 1 gexp 1 r pc
ð22Þ
gcom
1.8
Otto
Performance parameter
1.6 1.4
(S gen /c V)
1.2 1.0
(W net /c VT1 )
0.8 0.6 0.4
( ηth )
0.2 0.0 0
10
20 Compression ratio
1.8
40
Diesel
1.6
Performance parameter
30
(S gen /c V)
1.4 1.2 1.0
(W net /c VT1 )
0.8 0.6
( ηth )
0.4 0.2 0.0 0
10
20 Compression ratio
1.8
40
Brayton
1.6
Performance parameter
30
(S gen /c P)
1.4 1.2 1.0
(W net /c PT1 )
0.8 0.6
( ηth )
0.4 0.2 0.0 0
10
20 30 Pressure ratio
40
50
Fig. 2. Variation of thermal efficiency and dimensionless work output and dimensionless entropy production with compression ratio (Otto and Diesel)/ pressure ratio (Brayton).
Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
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where rp denotes the pressure ratio defined as rp = P2/P1. Likewise, with a similar approach explained earlier for the Otto cycle, the following expressions are resulted for the work output, thermal efficiency and entropy generation of the Brayton cycle.
" ðW net ÞBrayton ¼ cP T 1
gBrayton ¼
#
r ðc1Þ=c 1 p ð1cÞ=c ; rT gexp 1 rp
ð23Þ
gcom
rðc1Þ=c 1 ð1cÞ=c r T gexp 1 r p pg com
ðc1Þ=c
ðrT 1Þ
rp
1
ð24Þ
;
gcom
( ðSgen ÞBrayton ¼ cP
)
r ðc1Þ=c 1 p ð1cÞ=c ðr T 1Þð1 pÞ gexp r T 1 r p p : þ
gcom
ð25Þ
Also, the optimum pressure ratios leading to a maximum work output and a minimum entropy generation are c
ðr p ÞW max ¼ ðr T gexp gcom Þ2c2 ; ðr p ÞSgen; min ¼
r T
p
gexp gcom
2cc2
ð26Þ 2cc2 1 ¼ r p W max :
p
ð27Þ
The pressure ratio at which the net work output approaches zero, is c
c1 2 rp W net ¼0 ¼ rT gexp gcom ¼ r p W max :
ð28Þ
By comparing Eqs. (27) and (28), it can be inferred that r p S > rp W net ¼0 . Thus, a design based on minimum entropy gengen; min eration criterion would result in a negative net work output; a result that we obtained earlier for the Otto cycle. As in the case of Otto and Diesel cycles, a further conclusion is that the maximum work and minimum entropy production criteria lead to two different designs; see Eqs. (12), (19) and (27). 3. Illustrative numerical results A typical numerical example is illustrated in Fig. 2 where the thermal efficiency, dimensionless work output (Wnet divided by specific heat T1) and dimensionless entropy production (Sgen divided by specific heat) of each cycle are presented vs. system compression ratio. These results are obtained for rT = 4, gexp = 0.9, gcom = 0.85, p = 0.18, and c = 1.4. Note that for the Diesel cycle, the calculations are performed satisfying inequality 1 < rcut < r. Fig. 2 demonstrates that the entropy generation does neither correlate with the work output nor with the thermal efficiency. The entropy production monotonically decreases with the compression ratio but it attains a minimum value at a compression ratio much higher than the optimum compression ratios leading to a maximum work output and a maximum thermal efficiency. In all three engines, the work output and the efficiency are zero at the compression ratio of 1; beyond which both work output and thermal efficiency increase with the compression ratio until they attain maximum values; each at a certain compression ratio, but they decline with an increase in the compression ratio. As seen in Fig. 2, in any of three engines, both work output and thermal efficiency approach zero at an identical compression ratio. The work output and the thermal efficiency of the Otto, Diesel and Brayton cycles approach zero at a compression ratio of 16, 17.5 and 48, respectively. An interesting observation is that the optimum compression ratio corresponding to maximum work output is different from and less than the optimum compression ratio giving a maximum thermal efficiency. However, in more complex configurations such as regenerative conventional and hybrid gas turbine power plants, the maximum thermal efficiency occurs at a pressure ratio less than the one which gives a maximum power output [14]. 4. Discussion The results show that it is incorrect to interpret the production of entropy as a measure of losses in a heat engine, because as demonstrated, minimization of the production of entropy neither correlates with the optimization of thermal efficiency, nor with the maximization of work output. The best physical meaning of entropy production, as viewed and described by Leff [15] and Lambert [16,17], is that the generation of entropy is a measure of dispersal of energy, or the tendency of energy to spread out in space. Thus, application of a second law analysis for assessment of the design of a real power producing device may not provide conclusive design guidelines. It is however worth mentioning that the regime of entropy production may, at certain conditions, become equivalent to those of maximum efficiency and maximum work output. Leff and Jones [5] argue that such equivalence may happen if, for instance, one of the heat interactions QH and QL is kept constant. Bejan [18] presented specific models of endo-reversible
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
power plants which operate at maximum power output and at the same time produce minimum entropy generation. The models of power producing devices examined by Bejan [18] undergo isentropic compression and expansion processes. Another case is a regenerative gas turbine engine in which the operational regimes of minimum entropy production rate, maximum thermal efficiency and maximum power output are identical at a regenerative effectiveness of 50% [1]. Further useful results can be obtained by combination of the first law and the entropy generation equation; i.e. Eq. (7); this is the so-called Gouy-Stodola treatment. The work output is simply given by
W ¼ QH QL
ð29Þ
Eliminating QL between Eqs. (7) and (29), we find
TL Q H T L Sgen W ¼ 1 TH
ð30Þ
For known TL and TH, Eq. (30) shows that when the input heat is fixed, minimization of entropy generation is identical to maximization of work output. Dividing Eq. (30) by QH, we get another relationship as follows.
gth ¼ gC T L
Sgen QH
ð31Þ
where gC is the Carnot efficiency. Eq. (31) reveals that the thermal efficiency of an engine is proportional to the ratio of Sgen to QH. For a fixed QH, it is obvious that Max (gth) is equivalent to Min (Sgen). In such a specific condition, maximum work operation is also identical to the regime of maximum efficiency. In general case when QH is a variable, according to Eqs. (30) and (31), minimization of Sgen is not necessarily equivalent to maximization of work output, nor is it identical with maximization of thermal efficiency. These are the reasons why Min (Sgen) did neither correlate with Max (gth) nor with Max (W) in the irreversible power cycles examined in this paper. A question that may be raised is, if minimization of entropy production does not lead to an improvement of the performance of a heat engine, what would be the outcome of applying a second law analysis to a device in which the production of power is absent; e.g. a heat exchanger? Shah and Skiepko [19] and Fakheri [20] concluded that in a heat exchanger problem, minimization of irreversibility is not a useful tool for analyzing heat exchangers. In another work by Haseli et al. [21], the entropy generation of condensation of steam in the presence of air was studied in a shell and tube condenser. The results showed that a higher condensation rate would lead to a larger entropy production rate. If one were to follow the method of minimization of entropy production, he would need to decrease the rate of steam condensation, which would be irrational from the viewpoint of a heat exchanger designer. These observations and the results presented in this paper together with the critical arguments provided in Ref. [4–7] are expected to provide an opportunity to revisit our current view about entropy-based tools when designing thermal systems. 5. Conclusion Three most common types of heat engines including Otto, Diesel and Brayton cycles experiencing external and internal irreversibilities are examined to investigate the work output and the thermal efficiency of these engines at the regime of minimum entropy production. It is shown that an engine designed based on a minimum entropy production criterion may operate at a low efficiency, lower than maximum achievable efficiency. The main conclusion is that minimization of entropy generation is not necessarily equivalent to minimization of energy losses taking place in real engines. Because, the production of entropy is not a measure of disorder or losses in an energy system, rather, it is a measure of dispersal of energy, or the tendency of energy to spread out in space, as described by Leff [15] and Lambert [16,17]. It is also discussed that the operational regime of a heat engine at minimum entropy production may be identical with that of maximum efficiency and maximum work output under certain circumstances; e.g. fixed heat input. Acknowledgment I am grateful to the all reviewers for their useful and unbiased comments which allowed me to further improve the content of the paper. References [1] Y. Haseli, Optimization of a regenerative Brayton cycle by maximization of a newly defined second law efficiency, Energy Convers. Manage. 68 (2013) 133–140. [2] J.M.M. Roco, S. Velasco, A. Medina, A.C. Hernandez, Optimum performance of a regenerative Brayton thermal cycle, J. Appl. Phys. 82 (1997) 2735–2741. [3] A.C. Hernandez, A. Medina, J.M.M. Roco, Power and efficiency in a regenerative gas turbine cycle, J. Phys. D Appl. Phys. 28 (1995) 2020–2023. [4] P. Salamon, K.H. Hoffmann, S. Schubert, R.S. Berry, B. Andresen, What conditions make minimum entropy production equivalent to maximum power production, J. Non-Equilib. Thermodyn. 26 (2001) 73–83. [5] H.S. Leff, G.L. Jones, Irreversibility, entropy production, and thermal efficiency, Am. J. Phys. 43 (1975) 973–980. [6] P. Salamon, A. Nitzan, B. Andresen, R.S. Berry, Minimum entropy production and the optimization of heat engines, Phys. Rev. A 21 (1980) 2115–2129. [7] P. Salamon, A. Nitzan, Finite time optimizations of a Newton’s law Carnot cycle, J. Chem. Phys. 74 (1981) 3546–3560.
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[8] A. Bejan, The equivalence of maximum power and minimum entropy generation rate in the optimization of power plants, J. Energy Res. Technol. 118 (1996) 98–101. [9] F.L. Curzon, B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43 (1975) 22–24. [10] P. Chambadal, Le chix du cycle thermique dans une usine generatrice nucleaire, Revue Generate de l’Electricite 67 (1958) 332–345. [11] I.I. Novikov, The efficiency of atomic power stations, J. Nucl. Eng. 7 (1958) 125–128. [12] Y. Haseli. Thermodynamic optimization of power plants. Eindhoven University of Technology, Eindhoven, The Netherlands, 2011, ISBN: 978-9-03862522-5. [13] H.S. Leff, Thermal efficiency at maximum work output: new results for old heat engines, Am. J. Phys. 55 (1987) 602–610. [14] Y. Haseli, Optimum performance of a regenerative gas turbine power plant operating with/without a solid oxide fuel cell, J. Fuel Cell Sci. Technol. 8 (2011) 051003/1–051003/9. [15] H.S. Leff, Thermodynamic entropy: the spreading and sharing of energy, Am. J. Phys. 64 (1996) 1261–1271. [16] F.A. Lambert, Disorder-a crack crutch for supporting entropy discussions, J. Chem. Educ. 79 (2002) 187–192. [17] F.A. Lambert, Entropy is simple, qualitatively, J. Chem. Educ. 79 (2002) 1241–1246. [18] A. Bejan, Models of power plants that generate minimum entropy while operating at maximum power, Am. J. Phys. 64 (1996) 1054–1059. [19] R.K. Shah, T. Skiepko, Entropy generation extrema and their relationship with heat exchanger effectiveness – number of transfer unit behavior for complex flow arrangements, J. Heat Transfer 126 (2004) 994–1002. [20] A. Fakheri, Second law analysis of heat exchangers, J. Heat Transfer 132 (2010) 111802/1–111802/7. [21] Y. Haseli, I. Dincer, G.F. Naterer, Entropy generation of vapor condensation in the presence of a non-condensable gas in a shell and tube condenser, Int. J. Heat Mass Transfer 51 (2008) 1596–1602.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Short communication
Performance of irreversible heat engines at minimum entropy generation Y. Haseli ⇑ Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, Eindhoven 5600 MB, The Netherlands
a r t i c l e
i n f o
Article history: Received 24 August 2012 Received in revised form 21 May 2013 Accepted 23 May 2013 Available online 5 June 2013 Keywords: Heat engine Entropy generation Efficiency Work output Finite time Thermodynamics
a b s t r a c t A thermodynamic analysis is presented by means of mathematical formulation to examine the performance of the most common types of heat engines including Otto, Diesel, and Brayton cycles, at the regime of minimum entropy generation. All engines are subject to internal and external irreversibilities. It is shown that minimum entropy production criterion neither correlates with maximum thermal efficiency design nor with maximum work output criterion. The results demonstrate that the production of entropy is not necessarily equivalent to the energy losses taking place in real devices. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The topic of heat engines is an old subject of thermodynamics. The optimization of energy conversion systems; i.e. heat engines, has received remarkable attention by numerous scholars, in particular, since 1970s. Various optimization objectives have been employed to investigate the performance of well-known standard thermodynamic cycles including Brayton, Otto, Diesel, Atkinson, Miller, and Dual cycles. The most familiar optimization criteria examined in past works are maximum efficiency, maximum work (power) output, minimum entropy generation, and maximum ecological function (defined as power minus entropy generation rate times environment temperature). The methodology employed in past works for optimization of heat engines is based on Finite Time Thermodynamics (FTT), which suggests that to model a realistic operation of engines, it is necessary to account for finite time operation of the components of a given power cycle. This, unlike fully reversible cycles; e.g. Carnot, which consists of a sequence of infinitesimal processes, requires the various components of a power cycle to run in a finite time in order to produce a desired amount of power (work). The most practical optimization objectives in real life applications of power cycles are the thermal efficiency and the power output of a given engine. The design point corresponding to maximum efficiency is not necessarily identical to that of maximum power output. For instance, the maximum thermal efficiency and maximum work output for a regenerative Brayton-type heat engine are coincident only at a regenerative heat exchanger efficiency of 50% [1–3]. At the regime of maximum efficiency, we aim to extract as much work as possible per unit mass of fuel. On the other hand, at the regime of maximum work output, we aim to extract maximum possible power from engine without regard to how much fuel is spent. Whether the optimization should be performed according to maximum power or maximum efficiency depends on the problem constraints; the relative prices of power and fuel [4]. ⇑ Corresponding author. Present address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139, USA. E-mail addresses: [email protected], [email protected]. 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.05.010
Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
9811
The question of interest is when a heat engine is optimized based on a criterion other than maximum efficiency or maximum work output, what would be the efficiency and the net work production of the engine? In particular, if one designs an engine based on minimization of entropy production criterion, what would be the thermal efficiency and power output at the corresponding regime? To answer this question is the main objective of the present paper. Three well-known standard thermodynamic cycles including Brayton, Otto, and Diesel cycles which experiences internal and external irreversibilities are investigated to find whether minimization of entropy production associated with the operation of these engines may lead to conclusive design guidelines. The above question has somewhat been the interest of few authors. Indeed, in 1975, Leff and Jones [5] discussed by means of analytical argument that an increase in the thermal efficiency of an irreversible heat engine would not necessarily result in a decrease in its entropy production. Salamon et al. [6,7] argued in early 1980s that maximum work and minimum entropy production are two different designs in the optimization of heat engines. In 1996, Bejan [8] raised objection to the statement of Salamon et al. [6,7] arguing that according to Gouy–Stodola theorem maximum power output is equivalent to minimum entropy production rate. In his discussions, he pointed out the simplest and oldest power plants models of Curzon and Ahlborn [9], Chambadal [10] and Novikov [11], whose efficiency at maximum power operation is represented by gCA = 1 (TL/TH)½. Bejan argued that this efficiency could also be achieved by minimizing the entropy generation rate associated with the operation of the above models of power plants. Five years after Bejan’ criticism, Salamon et al. [4] argued through a comprehensive discussion that the justification of Bejan [8] about the equivalence of maximum work and minimum entropy production in heat engines is limited to only certain design conditions. An important observation made by Salamon et al. [4] was the method of evaluation of the total entropy production rate associated with the operation of a power producing compartment. In a recent book on power plants optimization [12], the power plant models of Curzon and Ahlborn [9], Novikov [11], modified Novikov [12], and Carnot vapor cycle is examined in detail. It is shown that the regime of minimum entropy production is different from that of maximum work output for the above first three models. For the case of a Carnot vapor cycle, which exchanges heat with a hot stream and a cold stream fluids through two heat exchangers at the hot-end and the codend sides of the engine, if the heat input is fixed and the energy contents of both hot and cold exhaust streams are not utilized in a secondary application, the operational regimes of minimum entropy production, maximum work output and maximum efficiency are identical. As mentioned previously, the central focus of the present paper is to examine the performance of three common power cycles; i.e. Brayton, Otto and Diesel, at the condition of minimum entropy production. In 1987, Leff [13] discovered that assuming constant specific heats, the efficiency of the standard air-cycles including Otto, Diesel, Joule-Brayton and Atkinson, at maximum work output could be represented exactly or approximately with Curzon–Ahlborn efficiency [9]. Leff had further assumed that the maximum and minimum temperatures of the above mentioned cycles were TH and TL, respectively. This paper aims to examine the performance of these heat engines at minimum entropy generation. The engines examined by Leff [13] and Cruzon and Ahlbon [9] are typical examples of endo-reversible cycles since the compression and expansion processes are assumed to be isentropic. In practical applications, however, it is almost unavoidable to neglect internal irreversibilities. A more realistic analysis would therefore require one to account for the irreversibilities of the compression and expansion processes together with the irreversible heat transfer processes between the engine and the high and low temperature reservoirs due to the finite temperature differences. Thus, we examine the performance of irreversible Otto, Diesel and Brayton cycles whose T–S diagram is depicted in Fig. 1. These engines undergo irreversible compression and expansion processes through lines 1–2 and 3–4, respectively. The isentropic compression and expansion processes are shown by dotted lines 1–2s and 3–4s, respectively. Line 2–3 represents a heat transfer process from the high temperature reservoir (TH) to the engine at constant volume/pressure/pressure in the Otto/Diesel/Brayton cycle. Line 4–1 shows the rejection of heat to the surrounding maintained at temperature TL at constant volume/volume/pressure in the Otto/Diesel/Brayton cycle. The isentropic efficiencies of the compression and the expansion processes are represented as gcom ¼ ðT 2s T 1 Þ=ðT 2 T 1 Þ and gexp ¼ ðT 3 T 4 Þ=ðT 3 T 4s Þ, respectively. Similar to the work of Leff [13], the analysis will be based on certain simplifying assumptions including, constant specific heat throughout the cycle, negligible mass of fuel compared to that of working gas, and an ideal gas behavior of the working substance (air). The maximum and minimum temperatures of the engine; i.e. T3 and T1, are the problem constraints due to the metallurgical limitation of the material of engines, and ambient conditions, since air is supplied from the atmosphere to the engines. 2. Mathematical formulation 2.1. Otto cycle Let us first consider the Otto cycle. As a primary step for evaluating the performance of the engine in terms of efficiency and work output, we need to determine temperatures T2 and T4 as functions of other operating parameters. Noting that ðT 3 =T 4s Þ ¼ ðT 2s =T 1 Þ ¼ r c1 , we get
T2 ¼ T1 þ
T 2s T 1
gcom
r c1 1 ; ¼ T1 1 þ
gcom
ð1Þ
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
TH
T
3
QH
4 4s 2 2s
QL 1
TL s Fig. 1. T–S diagram of irreversible Otto, Diesel and Brayton cycles.
h i T 4 ¼ T 3 gexp ðT 3 T 4s Þ ¼ T 3 1 gexp 1 r1c ;
ð2Þ
where r is the compression ratio defined as r = V1/V2 and c = cP/cV. The heat input and the heat output per unit mass of the air are obtained from
r c1 1 ; Q H ¼ c V ðT 3 T 2 Þ ¼ c V T 1 r T 1
ð3Þ
h i Q L ¼ cV ðT 4 T 1 Þ ¼ cV T 1 rT 1 gexp r T 1 r1c ;
ð4Þ
gcom
where rT = T3/T1. The work output and the thermal efficiency are therefore determined as follows.
r c1 1 ; ðW net ÞOtto ¼ Q H Q L ¼ cV T 1 rT gexp 1 r 1c
gcom
gOtto ¼
ðW net ÞOtto r T gexp gcom 1 r1c r c1 1 ¼ : QH ðrT 1Þgcom ðr c1 1Þ
ð5Þ
ð6Þ
Next, we evaluate the total entropy production associated with the operation of the engine. Entropy is generated due to four irreversible processes; i.e. the compression process (path 1–2), the heat transfer to the engine (path 2–3), the expansion process (path 3–4), and the heat rejection from the engine (path 4–1) to the ambient. Hence,
Sgen ¼
QL QH : TL TH
ð7Þ
Substituting Eqs. (3) and (4) into Eq. (7) and arranging yields
rc1 1 Sgen Otto ¼ cV ðr T 1Þð1 pÞ gexp rT 1 r 1c þ p ;
gcom
ð8Þ
where p = TL/TH. Notice that in the present analysis, TL = T1. To minimize the total entropy generation, we apply @Sgen =@r ¼ 0 whose solution gives
ðr ÞSgen; min ¼
r
T
p
gexp gcom
2c12
:
ð9Þ
We can also find an optimum compression ratio which would result in a maximum work output by applying ð@W net =@r Þ ¼ 0 to Eq. (5). This leads to
2c12 ðr ÞW max ¼ r T gexp gcom :
ð10Þ
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This results shows that the work output is zero at a certain compression ratio that is greater than ðr ÞW max ; which can be obtained by solving Eq. (5) with ðW net ÞOtto ¼ 0. Hence, 1
c1 ðr ÞW net ¼0 ¼ rT gexp gcom ¼ ðr Þ2W max :
ð11Þ
From Eqs. (9) and (10) it can be inferred that
ðr ÞSgen; min ¼ ðr ÞW max
2c12 1
p
ð12Þ
:
Equation (12) reveals that the operational regimes at maximum work output and minimum entropy generation are different. A further subtle observation can be made by comparing Eqs. (9) and (11): it can be readily shown that ðr ÞSgen; min > ðr ÞW net ¼0 . In other words, the work output of the engine at the regime of minimum entropy production would be negative! The maximum compression ratio that the engine may approach is ðrÞW net ¼0 . This argument leads us to conclude that it is desirable for the Otto cycle to operate at minimum entropy generation, and the efficiency at maximum work would be more desirable. 2.2. Diesel cycle For the case of the Diesel cycle, we undertake a similar procedure. We may still use Eq. (1) for determination of T2. However, T4 is obtained as follows.
(
"
T 4 ¼ T 3 1 gexp 1
r rcut
1c #) ;
ð13Þ
where 1 < rcut < r, and rcut denotes the cut-off ratio defined as rcut = V3/V2 > 1. It is related to the temperature ratio rT as
rT ¼ r c1 r cut :
ð14Þ
Consequently, we get the following expressions for the work output, the efficiency and the total entropy generation of the Diesel cycle.
(
"
ðW net ÞDiesel ¼ cV T 1 ðc 1Þðr T 1Þ þ gexp r T 1
r
1c #
r cut
r c1 1
gcom
)
c ;
ð15Þ
1c c1 r r g 1 c ðc 1Þðr T 1Þ þ gexp r T 1 rcut com
¼ ; r c1 1 c rT 1 g
gDiesel
ð16Þ
com
Sgen
Diesel
(
"
¼ cV ðrT 1Þð1 cpÞ gexp r T 1
r
r cut
1c # þ
r c1 1
gcom
)
cp :
ð17Þ
Maximization of the work output and minimization of the entropy generation result in the following optimum compression ratios.
21 ðr ÞW max ¼ r cT gexp gcom c 1 ;
ðr ÞSgen; min ¼
c rT
p
gexp gcom
ð18Þ
21
c 1
¼ ðr ÞW max
21 1 c 1
p
ð19Þ
:
As rcut > 1, a combination of Eqs. (14) and (19) leads to the following inequality.
rT >
gexp gcom ; p
ð20Þ
which implies that for any rT less than gexpgcom/p, minimization of entropy generation would lead to an rcut less than 1, which by definition is incorrect. For instance, assume gexp = 0.9, gcom = 0.85, and p = 0.18. For r T < 4:25ð¼ 0:9 0:85=0:18Þ, a design based on minimum entropy production criterion would require rcut < 1! 2.3. Brayton cycle c1
For the Brayton cycle, the temperatures T2 and T4 are obtained using the relationship ðT 3 =T 4s Þ ¼ ðT 2s =T 1 Þ ¼ r pc as follows.
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0
1 c1 c r 1 p A; T 2 ¼ T 1 @1 þ
ð21Þ
1c ; T 4 ¼ T 3 1 gexp 1 r pc
ð22Þ
gcom
1.8
Otto
Performance parameter
1.6 1.4
(S gen /c V)
1.2 1.0
(W net /c VT1 )
0.8 0.6 0.4
( ηth )
0.2 0.0 0
10
20 Compression ratio
1.8
40
Diesel
1.6
Performance parameter
30
(S gen /c V)
1.4 1.2 1.0
(W net /c VT1 )
0.8 0.6
( ηth )
0.4 0.2 0.0 0
10
20 Compression ratio
1.8
40
Brayton
1.6
Performance parameter
30
(S gen /c P)
1.4 1.2 1.0
(W net /c PT1 )
0.8 0.6
( ηth )
0.4 0.2 0.0 0
10
20 30 Pressure ratio
40
50
Fig. 2. Variation of thermal efficiency and dimensionless work output and dimensionless entropy production with compression ratio (Otto and Diesel)/ pressure ratio (Brayton).
Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
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where rp denotes the pressure ratio defined as rp = P2/P1. Likewise, with a similar approach explained earlier for the Otto cycle, the following expressions are resulted for the work output, thermal efficiency and entropy generation of the Brayton cycle.
" ðW net ÞBrayton ¼ cP T 1
gBrayton ¼
#
r ðc1Þ=c 1 p ð1cÞ=c ; rT gexp 1 rp
ð23Þ
gcom
rðc1Þ=c 1 ð1cÞ=c r T gexp 1 r p pg com
ðc1Þ=c
ðrT 1Þ
rp
1
ð24Þ
;
gcom
( ðSgen ÞBrayton ¼ cP
)
r ðc1Þ=c 1 p ð1cÞ=c ðr T 1Þð1 pÞ gexp r T 1 r p p : þ
gcom
ð25Þ
Also, the optimum pressure ratios leading to a maximum work output and a minimum entropy generation are c
ðr p ÞW max ¼ ðr T gexp gcom Þ2c2 ; ðr p ÞSgen; min ¼
r T
p
gexp gcom
2cc2
ð26Þ 2cc2 1 ¼ r p W max :
p
ð27Þ
The pressure ratio at which the net work output approaches zero, is c
c1 2 rp W net ¼0 ¼ rT gexp gcom ¼ r p W max :
ð28Þ
By comparing Eqs. (27) and (28), it can be inferred that r p S > rp W net ¼0 . Thus, a design based on minimum entropy gengen; min eration criterion would result in a negative net work output; a result that we obtained earlier for the Otto cycle. As in the case of Otto and Diesel cycles, a further conclusion is that the maximum work and minimum entropy production criteria lead to two different designs; see Eqs. (12), (19) and (27). 3. Illustrative numerical results A typical numerical example is illustrated in Fig. 2 where the thermal efficiency, dimensionless work output (Wnet divided by specific heat T1) and dimensionless entropy production (Sgen divided by specific heat) of each cycle are presented vs. system compression ratio. These results are obtained for rT = 4, gexp = 0.9, gcom = 0.85, p = 0.18, and c = 1.4. Note that for the Diesel cycle, the calculations are performed satisfying inequality 1 < rcut < r. Fig. 2 demonstrates that the entropy generation does neither correlate with the work output nor with the thermal efficiency. The entropy production monotonically decreases with the compression ratio but it attains a minimum value at a compression ratio much higher than the optimum compression ratios leading to a maximum work output and a maximum thermal efficiency. In all three engines, the work output and the efficiency are zero at the compression ratio of 1; beyond which both work output and thermal efficiency increase with the compression ratio until they attain maximum values; each at a certain compression ratio, but they decline with an increase in the compression ratio. As seen in Fig. 2, in any of three engines, both work output and thermal efficiency approach zero at an identical compression ratio. The work output and the thermal efficiency of the Otto, Diesel and Brayton cycles approach zero at a compression ratio of 16, 17.5 and 48, respectively. An interesting observation is that the optimum compression ratio corresponding to maximum work output is different from and less than the optimum compression ratio giving a maximum thermal efficiency. However, in more complex configurations such as regenerative conventional and hybrid gas turbine power plants, the maximum thermal efficiency occurs at a pressure ratio less than the one which gives a maximum power output [14]. 4. Discussion The results show that it is incorrect to interpret the production of entropy as a measure of losses in a heat engine, because as demonstrated, minimization of the production of entropy neither correlates with the optimization of thermal efficiency, nor with the maximization of work output. The best physical meaning of entropy production, as viewed and described by Leff [15] and Lambert [16,17], is that the generation of entropy is a measure of dispersal of energy, or the tendency of energy to spread out in space. Thus, application of a second law analysis for assessment of the design of a real power producing device may not provide conclusive design guidelines. It is however worth mentioning that the regime of entropy production may, at certain conditions, become equivalent to those of maximum efficiency and maximum work output. Leff and Jones [5] argue that such equivalence may happen if, for instance, one of the heat interactions QH and QL is kept constant. Bejan [18] presented specific models of endo-reversible
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Y. Haseli / Applied Mathematical Modelling 37 (2013) 9810–9817
power plants which operate at maximum power output and at the same time produce minimum entropy generation. The models of power producing devices examined by Bejan [18] undergo isentropic compression and expansion processes. Another case is a regenerative gas turbine engine in which the operational regimes of minimum entropy production rate, maximum thermal efficiency and maximum power output are identical at a regenerative effectiveness of 50% [1]. Further useful results can be obtained by combination of the first law and the entropy generation equation; i.e. Eq. (7); this is the so-called Gouy-Stodola treatment. The work output is simply given by
W ¼ QH QL
ð29Þ
Eliminating QL between Eqs. (7) and (29), we find
TL Q H T L Sgen W ¼ 1 TH
ð30Þ
For known TL and TH, Eq. (30) shows that when the input heat is fixed, minimization of entropy generation is identical to maximization of work output. Dividing Eq. (30) by QH, we get another relationship as follows.
gth ¼ gC T L
Sgen QH
ð31Þ
where gC is the Carnot efficiency. Eq. (31) reveals that the thermal efficiency of an engine is proportional to the ratio of Sgen to QH. For a fixed QH, it is obvious that Max (gth) is equivalent to Min (Sgen). In such a specific condition, maximum work operation is also identical to the regime of maximum efficiency. In general case when QH is a variable, according to Eqs. (30) and (31), minimization of Sgen is not necessarily equivalent to maximization of work output, nor is it identical with maximization of thermal efficiency. These are the reasons why Min (Sgen) did neither correlate with Max (gth) nor with Max (W) in the irreversible power cycles examined in this paper. A question that may be raised is, if minimization of entropy production does not lead to an improvement of the performance of a heat engine, what would be the outcome of applying a second law analysis to a device in which the production of power is absent; e.g. a heat exchanger? Shah and Skiepko [19] and Fakheri [20] concluded that in a heat exchanger problem, minimization of irreversibility is not a useful tool for analyzing heat exchangers. In another work by Haseli et al. [21], the entropy generation of condensation of steam in the presence of air was studied in a shell and tube condenser. The results showed that a higher condensation rate would lead to a larger entropy production rate. If one were to follow the method of minimization of entropy production, he would need to decrease the rate of steam condensation, which would be irrational from the viewpoint of a heat exchanger designer. These observations and the results presented in this paper together with the critical arguments provided in Ref. [4–7] are expected to provide an opportunity to revisit our current view about entropy-based tools when designing thermal systems. 5. Conclusion Three most common types of heat engines including Otto, Diesel and Brayton cycles experiencing external and internal irreversibilities are examined to investigate the work output and the thermal efficiency of these engines at the regime of minimum entropy production. It is shown that an engine designed based on a minimum entropy production criterion may operate at a low efficiency, lower than maximum achievable efficiency. The main conclusion is that minimization of entropy generation is not necessarily equivalent to minimization of energy losses taking place in real engines. Because, the production of entropy is not a measure of disorder or losses in an energy system, rather, it is a measure of dispersal of energy, or the tendency of energy to spread out in space, as described by Leff [15] and Lambert [16,17]. It is also discussed that the operational regime of a heat engine at minimum entropy production may be identical with that of maximum efficiency and maximum work output under certain circumstances; e.g. fixed heat input. Acknowledgment I am grateful to the all reviewers for their useful and unbiased comments which allowed me to further improve the content of the paper. References [1] Y. Haseli, Optimization of a regenerative Brayton cycle by maximization of a newly defined second law efficiency, Energy Convers. Manage. 68 (2013) 133–140. [2] J.M.M. Roco, S. Velasco, A. Medina, A.C. Hernandez, Optimum performance of a regenerative Brayton thermal cycle, J. Appl. Phys. 82 (1997) 2735–2741. [3] A.C. Hernandez, A. Medina, J.M.M. Roco, Power and efficiency in a regenerative gas turbine cycle, J. Phys. D Appl. Phys. 28 (1995) 2020–2023. [4] P. Salamon, K.H. Hoffmann, S. Schubert, R.S. Berry, B. Andresen, What conditions make minimum entropy production equivalent to maximum power production, J. Non-Equilib. Thermodyn. 26 (2001) 73–83. [5] H.S. Leff, G.L. Jones, Irreversibility, entropy production, and thermal efficiency, Am. J. Phys. 43 (1975) 973–980. [6] P. Salamon, A. Nitzan, B. Andresen, R.S. Berry, Minimum entropy production and the optimization of heat engines, Phys. Rev. A 21 (1980) 2115–2129. [7] P. Salamon, A. Nitzan, Finite time optimizations of a Newton’s law Carnot cycle, J. Chem. Phys. 74 (1981) 3546–3560.
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[8] A. Bejan, The equivalence of maximum power and minimum entropy generation rate in the optimization of power plants, J. Energy Res. Technol. 118 (1996) 98–101. [9] F.L. Curzon, B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43 (1975) 22–24. [10] P. Chambadal, Le chix du cycle thermique dans une usine generatrice nucleaire, Revue Generate de l’Electricite 67 (1958) 332–345. [11] I.I. Novikov, The efficiency of atomic power stations, J. Nucl. Eng. 7 (1958) 125–128. [12] Y. Haseli. Thermodynamic optimization of power plants. Eindhoven University of Technology, Eindhoven, The Netherlands, 2011, ISBN: 978-9-03862522-5. [13] H.S. Leff, Thermal efficiency at maximum work output: new results for old heat engines, Am. J. Phys. 55 (1987) 602–610. [14] Y. Haseli, Optimum performance of a regenerative gas turbine power plant operating with/without a solid oxide fuel cell, J. Fuel Cell Sci. Technol. 8 (2011) 051003/1–051003/9. [15] H.S. Leff, Thermodynamic entropy: the spreading and sharing of energy, Am. J. Phys. 64 (1996) 1261–1271. [16] F.A. Lambert, Disorder-a crack crutch for supporting entropy discussions, J. Chem. Educ. 79 (2002) 187–192. [17] F.A. Lambert, Entropy is simple, qualitatively, J. Chem. Educ. 79 (2002) 1241–1246. [18] A. Bejan, Models of power plants that generate minimum entropy while operating at maximum power, Am. J. Phys. 64 (1996) 1054–1059. [19] R.K. Shah, T. Skiepko, Entropy generation extrema and their relationship with heat exchanger effectiveness – number of transfer unit behavior for complex flow arrangements, J. Heat Transfer 126 (2004) 994–1002. [20] A. Fakheri, Second law analysis of heat exchangers, J. Heat Transfer 132 (2010) 111802/1–111802/7. [21] Y. Haseli, I. Dincer, G.F. Naterer, Entropy generation of vapor condensation in the presence of a non-condensable gas in a shell and tube condenser, Int. J. Heat Mass Transfer 51 (2008) 1596–1602.