International Journal of Heat and Mass Transfer 91 (2015) 150–161

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

A new approach to determining the intermediate temperatures ofendoreversible combined cycle power plant corresponding to maximumpower

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.0770017-9310/� 2015 Elsevier Ltd. All rights reserved.

E-mail address: jingwu12@gmail.com

Jing WuSchool of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 18 July 2014Received in revised form 20 July 2015Accepted 20 July 2015Available online 7 August 2015

Keywords:Endoreversible cycleFinite time thermodynamicsEntransy transfer efficiencyThermal efficiencyExergy

a b s t r a c t

Determining the optimal operating temperatures of power plants corresponding to maximum power isimportant for not only the analysis of cycle performance, but also the selection of appropriate workingfluids and their pressures. This study develops a new and convenient approach to determining the inter-mediate operating temperatures of n-stage endoreversible combined cycle power plants comprising n(arbitrary number) Carnot heat engines corresponding to the twice maximized power output by usingthe entransy transfer efficiency as an auxiliary parameter. The new approach reveals that when the tem-peratures of the hot and cold reservoirs, the total thermal conductance as well as the stage number of then-stage power plant are given, only two of these intermediate temperatures have fixed values, while theother ones are variable. It provides considerable flexibility for the designers to the selection of the opti-mal operating temperatures and appropriate working fluids. The procedures for determining all the pos-sible values of these intermediate temperatures are demonstrated. Next, a practical optimizationproblem of a two-stage combined cycle power plant is taken as an example to illustrate the superiorityof the newly proposed approach to the existing one. Finally, the physical meaning of entransy transferefficiency, together with its limitation is discussed and a comparison between the entransy-based effi-ciency and exergy-based efficiency is presented.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Improving the thermal performance of power plants has beenregarded as one of the key issues in energy conservation. It isknown that in all cycles between two heat reservoirs of differenttemperatures, the work output and the thermal efficiency are max-imal when the cycles are reversible. However, the power output ofa reversible cycle is zero since its operation time is infinitely longwhich is obviously meaningless for engineering applications. Inorder to obtain a certain amount of power, models of endore-versible cycles by considering the irreversibilities of finite-timeheat transfer processes are proposed by Chambadal [1], Novikov[2], and Curzon–Ahlborn (C–A) [3] and developed by Andersen,Salamon and Berry [4–6]. This method of modeling and optimizinga real thermodynamic cycle was referred to as finite time thermo-dynamics (FTT), a branch of thermodynamics devoted to extendclassical reversible thermodynamics to include more realistic pro-cesses. By now, this model has been widely used to analyze theperformance of heat engines, heat pumps and refrigerators [7–9]

with the main goal of ascertaining the performance bounds andoptimal criteria of selecting thermodynamic parameters of heatdevices with finite time cycles.

One of the endoreversible models that a number of researchersare interested in is the n-stage combined cycle power plant com-prising n (arbitrary number) reversible Carnot heat engines, asshown in Fig. 1 [10]. It is a universal model from which the optimalperformance concerning an arbitrary-stage endoreversible orreversible combined Carnot cycle system may be directly derived.Moreover, successful efforts have been made to use this model as areference one to analyze irreversible combined cycle powerplants by incorporating the most important irreversibilities into it[8,10–13].

Determining the intermediate operating temperatures of eachreversible cycle of the n-stage combined cycle power plant (i.e.,TH;i and TL;i in Fig. 1) corresponding to maximum power output isa major concern to engineers since they could select the appropri-ate working fluids and their operating pressures according to theoptimization result. When the number of stage, n, is small (n = 1or n = 2), there are only a few intermediate operating tempera-tures, and thus their values that correspond to maximum power

T

s

2

TL, 1

TL

TH, 2

Q1.

Q3.TL, 2

1

TH

Q2.

(UA)1

(UA)2

(UA)3

TH, 1

W1.

W2.

...

n

Qn+1.

(UA)n+1

Wn.

i

Qi+1.

(UA)i+1

Wi.

...

Qi.

(UA)i

Qn.

(UA)n

TH, i

TL, i

TH, n

TL, n

Fig. 1. An n-stage combined cycle power plant comprising n reversible Carnot heatengines, where each Carnot cycle in the system is connected through two heatexchangers and (UA)i is the thermal conductance of the ith heat exchanger, where Uis the heat transfer coefficient and A is the heat transfer area. Only the first and thelast heat engines are exposed to the hot (TH) and cold (TL) reservoirs, respectively[10].

TH TL

Qin.

Qout.

Fig. 2. Sketch of a heat transfer process.

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 151

can be readily obtained [14–16]. When the number of stage, n, islarger than two, however, the number of the intermediate temper-atures that are unknown and need to be determined may be quitelarge.

By now, several optimal operating conditions for the n-stagecombined cycle power plant have been reported. Bandyopadhyayet al. [10] have noted that all of the intermediate operating temper-atures corresponding to the maximum power may be chosen

arbitrarily, as long as the relationQn

i¼1TL;iTH;i¼

ffiffiffiffiTLTH

qis satisfied.

Furthermore, Bandyopadhyay et al. [10] and Bejan [16,17] havefound that the power produced by an n-stage combined powerplant can further be optimized for a given total thermal conduc-

tance (UA ¼Pnþ1

i¼1 ðUAÞi ¼ constant) when UA is divided equallyamong all the heat exchangers, that is, ðUAÞi ¼ UA=ðnþ 1Þ (the cor-responding power is referred to as twice maximized power outputin this paper, as in Refs. [17,18]). Herein, it may raise a question: Isthere any approach to determining the intermediate operatingtemperatures of an arbitrary-stage combined cycle power plantthat corresponds to the twice maximized power output? Obvi-

ously, the relationQn

i¼1TL;iTH;i¼

ffiffiffiffiTLTH

qalone cannot give a satisfactory

answer of this question since it can only tell us the relation ofthe intermediate operating temperatures corresponding to the‘‘once’’ maximized power output rather than the ‘‘twice’’ one. Tothe author’s best knowledge, this question has not been addressedby other existing literature either.

The purpose of this paper is to propose a convenient approachto determining the intermediate operating temperatures of anarbitrary-stage combined cycle power plant corresponding to thetwice maximized power output, and thereafter to show that whenTH, TL, n and UA are given, only two of these intermediate temper-atures have fixed values, while the other ones are variable, satisfy-ing a general formula. This study will help designers to make a

quicker judgment about the choice of working fluids and theiroperating pressures.

The plan of this paper is as follows. Considering that theentransy transfer efficiency serves as an auxiliary parameter foraccomplishing the purpose intended, we start this paper inSection 2 with a discussion of this concept from the aspects of itsorigin, definition and application. In Section 3, we introduce thenew approach and describe the procedures for finding all the pos-sible optimal intermediate temperatures. In Section 4, a two-stagecombined cycle power plant is taken as an example to illustrate thefeasibility and superiority of the newly proposed approach. Adiscussion of the physical meaning of entransy transfer efficiencyand a comparison between the entransy-based efficiency andexergy-based one are presented in Section 5. In Section 6, anattempt is made to derive the optimal intermediate temperaturesbased on entransy loss and entropy generation analysis. The paperconcludes with a summary and an appendix. All the processes/cycles discussed are assumed to operate continuously understeady-state conditions.

2. Entransy transfer efficiency

2.1. Origin

The essence of efficiency, in general, is the ratio of the producedvaluable resources to the consumed ones, reflecting how effec-tively the input is converted to the product. For instance, theheat-work conversion efficiency is defined as

Heat-work conversion efficiency ¼ Energy out in productEnergy in

¼_W

_Q in

:

ð1Þ

For a heat transfer process at steady-state, as shown in Fig. 2,however, if the product and consumable are quantified in the unitof heat (or heat flow), the heat transfer efficiency, quantitativelydetermined by the ratio of heat output to heat input, is always100% because of the energy conservation law, that is,

Heat transfer efficiency ¼ Energy out in productEnergy in

¼_Q out

_Q in

� 100%;

ð2Þ

which is meaningless to evaluate the performance of a heat transferprocess.

Recently, from the analogy between heat conduction and elec-tric conduction, Guo et al. [19,20] found that temperature, as thethermal potential, corresponds to the electrical potential, andFourier’s law corresponds to Ohm’s law. Therefore, the thermalenergy stored in an incompressible object should correspond tothe electrical charge stored in a capacitor. However, there is no

152 J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161

quantity in heat transfer theory corresponding to the electricalpotential energy of a capacitor. Hence, Guo et al. [19,20] defineda new physical quantity, G, to represents the heat transfer capabil-ity of an object during a time period. Its differential form isdG ¼ McV TdT and the integral form is G ¼ UT=2, where cV , T, M,U are respectively the constant volume specific heat capacity, tem-perature, mass and internal energy of an object and G is calledentransy, which was referred to as the heat transport potentialcapacity in an earlier paper [21]. Besides induction based on theanalogy between heat and electric conduction, the definition ofentransy can also be set up within the frame of continuummechanics [20] and it was found that the entransy is in fact asimplified expression for the potential energy of the phonon gas(thermomass) without the factor cV=c2 for convenience, where cis the speed of light in vacuum.

The entransy theory stresses the fact that the same quantity ofheat may have different quantities of entransy due to their differ-ent temperatures (i.e., thermal potentials), and hence heat can be‘‘weighted’’ according to its entransy. For instance, consider aone-dimensional heat transfer process through a finite tempera-ture difference, as shown in Fig. 3. During this process, heat isconserved due to the laws of energy conservation, but entransyis not conserved and dissipated.

For any (one-dimensional or multi-dimensional) heat transferprocess, the entransy balance equation can be expressed in rateform as [19,20]

_Gin|{z}Rate of total entransy entering the system

¼ _Gout|{z}Rate of total entransy leaving the system

þ _Gdiss|ffl{zffl}Rate of entransy dissipation

þ D _Gsystem|fflfflfflfflffl{zfflfflfflfflffl}Rate of change in the total entransy of the system

: ð3Þ

where _Gdiss P 0 ( _Gdiss ¼ 0 for reversible heat transfer processes and_Gdiss > 0 for irreversible ones). At steady state (D _Gsystem ¼ 0), Eq. (3)reduces to

_Gin|{z}Rate of total entransy entering the system

¼ _Gout|{z}Rate of total entransy leaving the system

þ _Gdiss|ffl{zffl}Rate of entransy dissipation

ð4Þ

Based on the concept of entransy dissipation, an optimizationprinciple, the entransy dissipation extremum principle (or theminimum entransy dissipation-based thermal resistance principle)

TH

TL

Qin.

Qout.

Wall

Gout.Gin

.

Entransydissipation

Fig. 3. Graphical representation of entransy dissipation during a one-dimensionalheat transfer process through a finite temperature difference.

has been developed and applied to some heat transfer optimizationproblems successfully (see the review paper by Chen et al. [20]).Moreover, Cheng and Liang et al. [22–24] extended the entransytheory to heat-work conversion processes and by developing theentransy balance equation for systems undergoing thermodynamicprocesses, they found that for a heat-work conversion cycle, thereason why the entransy flow at the heat flow outlet is smallerthan that at the inlet could be better explained by the concept ofentransy loss. Thereafter, the concept of entransy loss has beenapplied to analyzing and optimizing the thermodynamic processesor cycles by several research groups during the last 3 years[25–29].

2.2. Definition

As analyzed above, since heat is conserved during a heattransfer process, the heat transfer efficiency defined as the ratioof heat output to heat input (see Eq. (2)) is always 100%, whichis meaningless to evaluate the performance of a heat transferprocess. The non-conserved characteristic of entransy, however,provides a possible way to overcome this deficiency. We can definethe heat transfer efficiency as the ratio of the entransy out inproduct (the produced valuable resource) to the entransy in (theconsumed one), as [19]

gentransy ¼Entransy out in product

Entransy in

¼ 1� Entransy dissipationEntransy in

ð5Þ

or

gentransy ¼_Gout

_Gin

¼ 1�_Gdiss

_Gin

; ð6Þ

where gentransy is the heat transfer efficiency determined in terms ofentransy, which is referred to as the entransy transfer efficiency[19].

The entransy transfer efficiency can serve as a measure ofapproximation of an actual heat transfer process to the reversibleone and evaluate the heat transfer performance. Its value rangesbetween 0% and 100%. The lower limit of 0% corresponds to a com-plete dissipation in entransy, and the upper limit of 100% corre-sponds to the case of reversible heat transfer process through aninfinitely small temperature difference with no dissipation inentransy. For an actual heat transfer process through a finite tem-perature difference, the entransy-based efficiency is always lessthan 100% due to the always existing entransy dissipation.

In the case of one-dimensional heat transfer process withboundary temperatures TH and TL, as shown in Fig. 3, heat andentransy are entering from one side of the wall and leaving fromthe other side. The rates of entransy entering and leaving the sys-tem are _Gin ¼ _Q inTH and _Gout ¼ _Q outTL, respectively. Substituting

them into Eq. (6) with consideration of _Q in ¼ _Q out ¼ _Q , we canget the expression of entransy transfer efficiency of theone-dimensional heat transfer process, that is,

gentransy ¼_Gout

_Gin

¼_Q outTL

_Q inTH

¼_QTL

_QTH

¼ TL

TH: ð7Þ

It cannot exceed 100% for an actual process because of TL < TH,and becomes 100% under reversible conditions where TL ¼ TH.

2.3. Application to the analysis of some endoreversible models

For simplicity, consider first a simple endoreversible model, theChambadal’s model [1], operating between the two thermal energy

T

s

1

2 3

4

TH

TL

THCQH.

QL.

W.

ηentransy

η

1

0.7

0.5

0.4

(a) (b)

Fig. 4. Example of application of entransy transfer efficiency: (a) a T–s diagram ofChambadal’s model [1]; (b) variation of the thermal efficiency of Chambadal’smodel with the entransy transfer efficiency of the heat transfer process involved inthis model (assume that TL = 300 K and TH = 1000 K).

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 153

reservoirs at temperatures TH and TL, as shown in Fig. 4(a). Thismodel involves only one heat transfer process with boundarytemperatures TH and THC and only one Carnot engine with1-2-3-4-1 cycle that operates between THC and TL.

The thermal efficiency of the Chambadal’s model is

g ¼_W

_Q H

¼ 1�_Q L

_Q H

¼ 1� TL

THC: ð8Þ

This equation only refers to the upper and lower temperaturebounds of the reversible Carnot cycle (i.e., THC and TL) and is notin a convenient form for expressing explicitly how the heat transferperformance of the irreversible heat transfer process between TH

and THC involved in this model affects its thermal efficiency, g.Thus, it is desirable to rewrite Eq. (8) as

g ¼ 1� TL

THC¼ 1� TL

TH� 1gentransy

; ð9Þ

where gentransy is the entransy transfer efficiency of the heat transferprocess between TH and THC with the expression gentransy ¼ THC=TH,measuring its heat transfer performance.

The combination form containing TH, TL and gentransy as shown inEq. (9) reveals that the thermal efficiency of the Chambadal’smodel is influenced by two kinds of efficiencies: the Carnot effi-ciency between the two thermal energy reservoirs at TH and TL,gC ¼ 1� TL=TH, and the entransy transfer efficiency of the heattransfer process between TH and THC, gentransy. The Carnot efficiencyreflects the highest degree of heat that can be converted into workby a heat engine operating between TH and TL (i.e., the upper andlower temperature bounds of the Chambadal’s model). Theentransy transfer efficiency gauges the degree of approximationof the actual heat transfer process between TH and THC involvedin the Chambadal’s model to the corresponding reversible one(TH ¼ THC). It is just the imperfection originating from the finitetime heat transfer measured by the entransy transfer efficiency,gentransy, that causes the thermal efficiency of the Chambadal’smodel, g, to be less than that of the corresponding Carnot efficiencyoperating between the same thermal energy reservoirs at temper-atures TH and TL. For a totally reversible condition (that is, the heattransfer is through an infinitely small temperature difference), wehave gentransy ¼ 1 and then g ¼ 1� TL

THaccording to Eq. (9).

From Eq. (9), we can see that the higher the Carnot efficiency,gC ¼ 1� TL=TH, and entransy transfer efficiency, gentransy, the higherthe thermal efficiency, g. But in many engineering cases, TL is anexisting ambient temperature and TH is limited by the metallurgi-cal strength of available materials. Consequently, increasing theCarnot efficiency, 1� TL=TH, for the sake of a higher thermal

efficiency, g, is usually restricted in reality. By contrast, increasingthe entransy transfer efficiency, gentransy, by the method of heattransfer optimization is more realistic. For instance, optimizinga heat transfer process for prescribed heat flux boundary condi-tions may result in a minimum difference between the twoboundary temperatures (i.e., TH and THC) and may thus lead to ahigher entransy transfer efficiency and a resulting higher thermalefficiency.

Fig. 4(b) illustrates the variation of the thermal efficiency, g,with the entransy transfer efficiency, gentransy, of the Chambadal’smodel for a given values of TL ¼ 300 K and TH ¼ 1000 K, showingthat the thermal efficiency increases monotonically with theincreasing entransy transfer efficiency of the heat transfer processinvolved in this model.

To foster a deeper understanding of how the heat transferperformances of the irreversible heat transfer processes involvedin an endoreversible model affect its thermal efficiency (or COP),the expressions of thermal efficiency (or COP) of some other typesof endoreversible models [8,30,31] in terms of TL, TH and entransytransfer efficiency of each heat transfer process involved in thesemodels are given in Table 1. The derivation of these expressionsis discussed in detail in Appendix A.

3. New approach to determining the optimal operatingtemperatures

According to Eq. (7), the entransy transfer efficiency of the heattransfer process occurring in the ith heat exchanger of the n-stagecombined cycle power plant (Fig. 1) is

gentransy;i ¼TH;i

TL;i�1; ð10Þ

where i = 1, 2, . . . ,n + 1 (n P 1), TL;0 ¼ TH and TH;nþ1 ¼ TL.On the other hand, the overall thermal efficiency of the n-stage

power plant, g, is given by

g ¼ 1�_Q nþ1

_Q1

ð11Þ

and for each reversible heat engine, the second law of thermody-namic gives

_Q i

_Qiþ1

¼ TH;i

TL;i: ð12Þ

Combining Eqs. (10)–(12) yields

g ¼ 1� TL

TH� 1Qnþ1

i¼1 gentransy;i

; ð13Þ

which shows that the overall thermal efficiency can be expressed interms of TL, TH and the product of entransy transfer efficiencies ofeach heat transfer process occurring in the n-stage power plant.

Bandyopadhyay et al. [10] has found that the overall thermalefficiency of the n-stage combined power plant is g ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiTL=TH

pwhen it delivers maximized power output with the linear heat trans-fer law, independent of the stage number n. An equivalent result forthe two-stage combined cycle power plant has been reported byRubin et al. [14] and Bejan [16]. Substituting this result into Eq.(13), the product of the entransy transfer efficiencies of each heattransfer process at maximized power output is

Ynþ1

i¼1

gentransy;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiTL=TH

p; ð14Þ

As mentioned in the introduction part, for a given total thermal

conductance, UA ¼Pnþ1

i¼1 ðUAÞi, the twice maximized power outputis obtained when UA is divided equally among all the heat

Table 1Thermal efficiencies (or COPs) of some types of endoreversible cycle expressed in terms of TL, TH and entransy transfer efficiency.

Model T–s diagram Expression of thermal efficiency/COP

Novikov’s model T

s

1

2 3

4

TH

TL

THC

.4irr

QH.

QL.

(UA)H

g ¼ 1� ð1þ iÞ TLTH� 1gentransy

Carnot cycle with finite heat reservoirs T

s

TH,out

THC

QH.

TLC

QL.

TL

TL,out

TH

1

2 3

4

g ¼ 1� TLTH

2gentransy;I f H

� gH

� �� 2gentransy;II

f L� gL

� ��1

Brayton cycle with finite heat reservoirs T

s

TH,out QH.

QL.

TL

TL,out

TH

1

2

3

4

g ¼ 1� TLTH�

CLCb�1

CHCa�1

!�

CHCa

1gentransy;I

�1CLCb

gentransy;II�1

!

Stirling cycle with finite heat reservoirs T

s

TH,out

THSQH.

TLS

QL.

TL

TL,out

TH

QR.

12

3 4

ΔQR.

g ¼1�TL

TH2

gentransy;I f H�gH

h i�

2gentransy;IIf L

�gL

h i�1

1þ1�gentransy;Rð ÞCR 1�eRð Þ

_mRg lnðVmax=Vmin Þ

Reversed Carnot refrigeration cycle T

s

TH

TL

THC

QH.

QL.

TLC

W.

COP ¼ 11

gentransy;I� 1gentransy;II

�THTL�1

Reversed Carnot heat pump cycle COP ¼ 11�gentransy;I �gentransy;II �

TLTH

154 J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161

exchangers, i.e., ðUAÞi ¼ UA=ðnþ 1Þ [10,16,17]. By combiningthe expression of the heat transfer rate through the ith heat

exchanger, _Qi ¼ ðUAÞiðTL;i�1 � TH;iÞ, with Eqs. (10) and (12) andðUAÞi ¼ UA=ðnþ 1Þ yields

1gentransy;i

þ gentransy;iþ1 ¼ 2: ð15Þ

So far, we have obtained two equations, Eqs. (14) and (15), thatrelate the entransy transfer efficiencies of each heat transfer pro-cess involved in the n-stage combined cycle power plant corre-sponding to the twice maximized power output on the basis ofthe optimization results reported in previous literature[10,16,17]. Combining these two equations and solving for theentransy transfer efficiency of the heat transfer process occurringin the ith heat exchanger, we finally get

gentransy;i ¼nþ 1� iþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiTL=TH

p=Qi�1

k¼1gentransy;k

nþ 2� i: ð16Þ

Note that in order to make Eq. (16) in an elegant manner,Q0k¼1gentransy;k is assumed to be unity without attaching any

physical meaning to it.Based on Eq. (16), all the possible intermediate operating tem-

peratures of an n-stage combined cycle power plant correspondingto the twice maximized power output for given TH, TL, UA and n,can be determined in a step-by-step manner as follows:

(1) Substitute TH, TL and n into Eq. (16) and calculate gentransy;i

(¼ TH;iTL;i�1

) for i = 1 to i = n + 1 in sequence,

(2) Determine TH;1 and TL;n from the ratios gentransy;1 ¼ TH;1=TH

and gentransy;nþ1 ¼ TL=TL;n,

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 155

(3) Choose arbitrarily other intermediate temperaturesbetween TH and TL as long as the ratios obtained by step(1) and the relation TL;iþ1 < TH;iþ1 < TL;i < TH;i are satisfiedsimultaneously.

A few observations can be made from the procedures above.First, the concept of entransy transfer efficiency serves an auxiliaryparameter for the determination of all the possible intermediateoperating temperatures.

Another observation that can be made from the proceduresabove is that two of the intermediate temperatures, TH;1 and TL;n,corresponding to the twice maximized power output are fixed tocertain values, while all the other ones are variable. To illustrateit more clearly, let us consider a four-stage power cycle withTH ¼ 1200 K and TL ¼ 300 K as an example, which involves fiveheat transfer processes. Substituting TH ¼ 1200 K and TL ¼ 300 K

as well as n = 4 into Eq. (16) gives gentransy;1 ¼TH;1TH¼ 9

10 and

gentransy;5 ¼TL

TL;4¼ 5

6. Then we obtain TH;1 ¼ 1080 K and TL;4 ¼ 360 K

from these two ratios, respectively, showing that TH;1 and TL;4 are

both fixed. Moreover, we have gentransy;2 ¼TH;2TL;1¼ 8

9, gentransy;3 ¼TH;3TL;2¼ 7

8,

gentransy;4 ¼TH;4TL;3¼ 6

7 using Eq. (16). The temperatures, TH;4, TL;3, TH;3,

TL;2, TH;2 and TL;1 can vary arbitrarily as long as these three ratiosand the relation 360 K < TH;4 < TL;3 < TH;3 < TL;2 <

TH;2 < TL;1 < 1080 K are satisfied simultaneously, which meansthat designers have enough flexibility in selecting the appropriateworking fluids and their operating pressures.

In addition, the twice maximized power output of the n-stagecombined cycle power plant, _W total, with given TH, TL, n and UA,can be expressed as

_W total ¼Xn

i¼1

_Wi ¼ _Q 1 � _Q nþ1 ¼ ðUAÞ1ðTH � TH;1Þ � ðUAÞnþ1ðTL;n � TLÞ

¼ UAnþ1

ðTH � TH;1Þ� TL;n � TLð Þ½ �; ð17Þ

where TH;1 and TL;n are fixed values according to the analysis above.Then, we can conclude from Eq. (17) that the n-stage power plantwill generate the same amount of the twice maximized poweroutput with given TH, TL, n and UA, independent of the values of

Table 2Comparison of the optimization process of the newly proposed approach and the existing(TH = 1200 K, TL = 300 K and (UA)1 + (UA)2 + (UA)3 = 3 W/K).

Approach I [16]

Illustration ofoptimizationprocess

TL1

TL

TH2

QH.

QL.TL2

TH

QM.

TH1

W1.

W2.

Cycle 1

Cycle 2

C1

First step

(Two steps)

Secondstep

Optimization results TH1 (K) 1000TL1 (K) 2500/3TH2 (K) 2000/3TL2 (K) 400_W1 (W) 100/3_W2 (W) 200/3_W1 þ _W2 (W) 100

the individual intermediate temperature selected as long as theysatisfy Eq. (16).

4. Case study and discussion

Consider a two-stage combined cycle power plant. AssumingTH ¼ 1200 K, TL ¼ 300 K, and the total thermal conductance,ðUAÞ1 þ ðUAÞ2 þ ðUAÞ3, to be 3 W=K. A comparison between thenewly proposed approach and the existing ones [16] are shownin Table 2.

Approach I is a two-step approach proposed in Ref. [16]. In thefirst step, the two-stage power plant is divided into two cycles: oneis contained between TH and TL1 (cycle 1), the other is containedbetween TL1 and TL (cycle 2). The optimizations of cycles 1 and 2are executed separately to obtain the temperature relations corre-sponding to the maximum _W1 and _W2 for cycles 1 and 2, respec-tively. In the second step, _W1 þ _W2 is further (twice) maximized.

Inspired by the ideas of approach I [16], approach II could benaturally developed here, which is also a two-step approach. Butnote that this time the two-stage power plant is divided into twocycles in a different way: one is contained between TH and TH;2

(cycle 10), the other is contained between TH;2 and TL (cycle 20). In

the first step, _W1 and _W2 for cycles 10 and 20 are optimized respec-tively, and in the second step, _W1 þ _W2 is further maximized. Theoptimization results by approaches I and II are listed in Table 2.

Approach III is just the one newly proposed in this study. First,substituting the known values of TH ¼ 1200 K, TL ¼ 300 K and

n ¼ 2 into the general formula, Eq. (16), gives gentransy;1 ¼TH;1TH¼ 5

6,

gentransy;2 ¼TH;2TL;1¼ 4

5 and gentransy;3 ¼TL

TL;2¼ 3

4. Next, TH;1 ¼ 1000 K and

TL;2 ¼ 400 K are determined from TH;1TH¼ 5

6 and TLTL;2¼ 3

4, respectively.

But unlike the temperatures, TH;1 and TL;2, which have fixed values,the values of temperatures, TH;2 and TL;1, could be chosen arbitrar-

ily as long as the ratio TH;2TL;1¼ 4

5 and the limitation

400 K < TH;2 < TL;1 < 1000 K are satisfied simultaneously. As a

result, the power output of the two Carnot cycles, _W1 and _W2, cor-responding to the twice maximized power output are also variable,as given in the last column of Table 2.

ones and the optimization result corresponding to the twice maximized power plant

II III (this study)

ycles &2

Cycle 2'

Cycle 1'TL1

TL

TH2

QH.

QL.TL2

TH

QM.

TH1

W1.

W2.

Cycles 1'&2'

First step

(Two steps)

Secondstep

TL1

TL

TH2

QH.

QL.TL2

TH

QM.

TH1W1.

W2.

General formula (eq. (16))

1000 10002000/3 x (variable)1600/3 0.8x (variable)400 400200/3 200 � 0.2x (variable)

100/3 0.2x � 100 (variable)

100 100

Fig. 5. The optimal temperatures TH,2 and TL,1 determined by approaches I and II arefixed values (illustrated by two points), while those determined by approach III arevariable that meet the relation TH;2 ¼ 4

5 TL;1 and 400 K < TH;2 < TL;1 < 1000 K simul-taneously (illustrated by the line segment).

156 J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161

Approaches I and II give, respectively, only one choice of thepossible intermediate temperatures corresponding to the twicemaximized power output, and one is not necessarily better thanthe other. Approach III, however, appears to be more efficientand convenient since all the possible choices of the intermediatetemperatures can be obtained directly from Eq. (16). In fact, theoptimal temperatures, TH;2 and TL;1, determined by approach I orapproach II, as shown in Table 2, is only one of the possible choicesdetermined by approach III, as illustrated in Fig. 5.

Note that the amount of the twice maximized power output,_W1 þ _W2, is always the same (100 W for this example), as shown

in Table 2, independent of the values of the individual intermediatetemperature selected, verifying the analysis in the last paragraphof Section 3.

5. A comparison between entransy-based efficiency and exergy-based efficiency

Engineers make frequent use of exergy-based efficiency togauge the heat transfer performance with the idea that exergy isnot conserved during irreversible heat transfer processes. Theexpression is [32]

gexergy ¼Exergy outExergy in

¼ 1� Exergy destructionExergy in

; ð18Þ

which is also referred to as the second law efficiency.This definition of the exergy-based efficiency, gexergy, has an

analogous form as that of the entransy-based efficiency, gentransy,as shown in Eq. (5). Besides, both the exergy-based efficiencyand entransy-based efficiency can serve as the performanceparameters of heat transfer processes to measure the approxima-tion of an actual heat transfer process to the reversible one. Butwhen we attempt to use the exergy-based efficiency as an auxiliaryparameter to determine the optimal intermediate operating tem-peratures of n-stage combined power plants (as what we havedone by using the entransy-based efficiency in Section 3), it isfound that establishing the explicit equations, like Eqs. (14) and(15), that relate the unknown intermediate temperatures (i.e., TH;i

and TL;i) and the known values of TH, TL and n with the help ofthe exergy-based efficiency is considerably difficult, especiallywhen the number of stage, n, is larger than two. The reason is thataccording to Eq. (18), the exergy-based efficiency of the heattransfer process occurring in the ith heat exchanger of the n-stagecombined cycle power plant is

gexergy;i ¼Exergy outExergy in

¼ 1� T0=TH;i

1� T0=TL;i�1; ð19Þ

which is much more complicated than the expression of theentransy-based efficiency as shown in Eq. (10). Therefore, in thisstudy, we take the entransy-based efficiency instead of theexergy-based one as the auxiliary parameter for the determinationof all the possible intermediate operating temperatures correspond-ing to the twice maximized power output.

Furthermore, the physical meanings of the entransy-based effi-ciency and exergy-based efficiency of a heat transfer process arecompared, as summarized in Table 3. We know that exergy (whichis also called the availability or available energy) is a measure ofthe useful work potential of energy [33]. Thus, in the definitionof exergy-based efficiency, Eq. (18), energy is ‘‘weighted’’ accordingto its useful work potential and the exergy-based efficiency mea-sures the losses in work capability during heat transfer processes[34]. If we investigate how well availability is used during the heattransfer processes, it may be more appropriate to express this kindof heat transfer performance by the exergy-based efficiency. In thedefinition of entransy-based efficiency, Eq. (5), however, energy is‘‘weighted’’ according to its entransy (i.e., heat transfer capability[19,20]), and thus entransy transfer efficiency accounts for the dis-sipations in heat transfer capability during heat transfer processes.If what we are concerned with is how well heat is transferredduring the heat transfer processes, this kind of heat transfer perfor-mance could be better characterized by the entransy-basedefficiency.

Reconsider the one-dimensional heat transfer process through afinite temperature difference as an example, as shown in Table 3.Using Eq. (18), the exergy-based efficiency of this process is

gexergy ¼Exergy outExergy in

¼ 1� T0=TL

1� T0=TH; ð20Þ

which conveys an important message that is equally applicable toboth one-dimensional and multi-dimensional heat transfer pro-cesses: the value of gexergy is influenced not only by the tempera-tures of the heat-transfer medium (TH and TL) but also by thetemperature of the environment (T0). It means that if we took theexergy-based efficiency as the performance parameter to assesshow well heat is transferred during the transfer process with givenboundary temperatures of TH and TL, a conclusion would be reachedfrom Eq. (20) that the heat transfer performance of this process inwinter is better than that in summer (see the sketch in the last col-umn of Table 3) since the temperature of the environment, T0, inwinter is usually much lower than that in summer. But there is acommon-sense understanding that how well heat is transferredfor a heat transfer process depends on the conditions of theheat-transfer medium only, and not on that of the environment.Thus, from this example we can see that the exergy-based efficiencymay be not applicable to evaluate how well heat is transferredduring a heat transfer process. On the other hand, the entransy-based efficiency of this one-dimensional heat transfer process isgentransy ¼

TLTH

. That is, the entransy-based efficiency depends on the

temperatures of heat-transfer medium only, making no referenceto T0. It means that the entransy-based efficiency has a fixed valueif the boundary temperatures, TH and TL, of the heat transfer processare given (see the sketch in the second column of Table 3). Hence,the entransy-based efficiency may be more appropriate to evaluatehow well heat is transferred during a heat transfer process.

The non-zero reference point is the reason for the environmen-tal temperature-dependent characteristic of the exergy-based effi-ciency. As we known, the reference point of exergy is the state inthermodynamic equilibrium with the environment rather thanthe absolute zero temperature. In contrast, the reference point of

Table 3A comparison between the expressions and physical meanings of the entransy-based efficiency and exergy-based efficiency of a heat transfer process.

Entransy-based efficiency (referred to as entransy transfer efficiency) Exergy-based efficiency (referred to as second-law efficiency)

Expression gentransy ¼Entransy outEntransy in ¼ 1� Entransy dissipation

Entransy in gexergy ¼Exergy outExergy in ¼ 1� Exergy destruction

Exergy in

Physical meaning A measure of dissipations in heat transfer capability during a heattransfer process; evaluates how well heat is transferred during thisprocess

A measure of losses in work capability during a heat transfer process;evaluates how effectively availability is used during this process

TH TL

Qin.

Qout.

Wall

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 157

entransy is the absolute zero temperature (see Eq. (2) in Ref. [19]),and as a result the entransy-based efficiency is independent of T0.

Despite its successful application in determining the intermedi-ate temperatures in this study, the concept of entransy transferefficiency still has its limitation. It is clear from its definition,gentransy ¼ 1� Entransy dissipation

Entransy in , that the non-unity entransy transfer

efficiency for an actual heat transfer process is essentially due tothe non-zero entransy dissipation rate during this process. Basedon the entransy balance equation, the expression of the entransy

dissipation rate, kðrTÞ2, can be obtained (discussed in detail else-where [19,20]), which indicates that the entransy dissipation ratedepends only on the temperature gradient,rT , as well as the ther-mal conductivity, k. That is, only the imperfection originating fromthe finite temperature difference heat transfer can be accountedfor by the entransy transfer efficiency. Other imperfections thatmay cause an actual process to deviate from the idealized one, suchas friction or unrestrained expansion during a work transfer pro-cess, however, cannot be gauged by this concept.

6. An attempt to derive the optimal intermediate temperaturesbased on entransy loss and entropy generation analysis

It is noted that Cheng et al. [35] applied the concept of entransyloss to the optimization of a two-stage combined cycle powerplant. Following the expressions of the total entransy loss rateand entransy loss coefficient that have been obtained in their work(see Eqs. (14) and (15) in Ref. [35]), for the n-stage combined cyclepower plant as shown in Fig. 1, we have

_Gloss ¼ _Q 1TH � _Q nþ1TL ð21Þ

and

gG ¼_Gloss

_Gin

¼_Q 1TH � _Q nþ1TL

_Q 1TH

¼ 1�_Qnþ1TL

_Q 1TH

; ð22Þ

where _Gloss is the total entransy loss rate during the n-stage com-bined cycle power plant and gG is the entransy loss coefficient ofthe combined cycle power plant. Substituting the relations_Q1 ¼ ðUAÞ1ðTH � TH;1Þ and _Qnþ1 ¼ ðUAÞnþ1ðTL;n � TLÞ into Eqs. (21)

and (22) yields

_Gloss ¼ ðUAÞ1 T2H � TH;1TH

� �� ðUAÞnþ1 TL;nTL � T2

L

� �ð23Þ

and

gG ¼ 1�ðUAÞnþ1 TL;n � TLð ÞTL

ðUAÞ1 TH � TH;1ð ÞTH: ð24Þ

Obviously, neither the total entransy loss rate expressed by Eq.(23) nor the entransy loss coefficient expressed by Eq. (24) can beused to determine the optimal intermediate operating tempera-tures of the n-stage combined cycle since the intermediate operat-ing temperatures, TH;i and TL;i (1 < i < n), are not included in thesetwo expressions at all. For the same reason, the total entropy gen-

eration rate of the n-stage combined cycle, _Sgen, with the expres-sion as

_Sgen ¼_Qnþ1

TL�

_Q 1

TH¼ðUAÞnþ1

TLTL;n � TLð Þ � ðUAÞ1

THTH � TH;1ð Þ; ð25Þ

cannot be used to determine the intermediate operating tempera-tures either.

In fact, for the application of total entransy loss rate or entropygeneration rate, it has been shown that larger total entransy lossrate or smaller total entropy generation rate leads to larger total

power output when the preconditions (i.e., TH, TL and _Q1 are fixed)are satisfied [35]. For the discussed n-stage combined cycle, _Q1 isnot fixed and thus the preconditions are not satisfied. This is thereason why the concept of total entransy loss rate or entropy gen-eration rate is not applicable to this problem. Moreover, for theapplication of entransy loss coefficient, it has been shown [35] thatwhen TL and TH are fixed, the larger entransy loss coefficient, thelarger thermal efficiency will be since the entransy loss coefficient,gG, shown in Eq. (22) can be rewritten in terms of TL, TH and thethermal efficiency, g, as

gG ¼ 1�_Q nþ1TL

_Q 1TH

¼ 1� TL

TH

_Q1 � _W total

_Q 1

!¼ 1� TL

THð1� gÞ: ð26Þ

But notice that the optimization objective of the discussed problemin this paper is the maximized power output rather than the max-imized thermal efficiency. Therefore, the concept of entransy losscoefficient is not applicable to the discussed problem.

Now we apply the concept of entransy loss or entropy genera-tion to each Carnot engine involved in the n-stage combined cyclein an attempt to obtain the optimal intermediate temperatures interms of the entransy loss extremum or entropy generation mini-mization. For simplicity, we assume that n = 2. Following the stud-ies in Ref. [16], we divide the two-stage power plant into twocycles: one is contained between TH and TL1 (cycle 1), the otheris contained between TL1 and TL (cycle 2) (see the figure in the sec-ond column of Table 2) and the temperature level TL1 is assumed tobe fixed at first, as in Ref. [16].

The entransy loss rate and entropy generation rate during theoperation of cycle 1 are respectively

158 J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161

_Gloss;cycle 1 ¼ _QHTH � _Q MTL1 ð27Þ

and

_Sgen;cycle 1 ¼ _Q Hð1

TH1� 1

THÞ: ð28Þ

Substituting the relations _QH= _Q M ¼ TH1=TL1 and_QH ¼ ðUAÞHðTH � TH1Þ into Eqs. (27) and (28) gives

_Gloss;cycle 1 ¼ ðUAÞHðTH � TH1ÞðTH �T2

L1

TH1Þ ð29Þ

and

_Sgen;cycle 1 ¼ ðUAÞHTH

TH1þ TH1

TH� 2

� �: ð30Þ

By solving @ _Gloss;cycle 1=@TH1 ¼ 0 and @ _Sgen;cycle 1=@TH1 ¼ 0, we getTH1 ¼ TL1 and TH1 ¼ TH, respectively. Obviously, these results aredifferent from the optimal results that obtained by Ref. [16] orby the newly proposed approach. It verifies again that the entransyloss or the entropy generation is not applicable to the determina-tion of the optimal intermediate temperatures, and then the opti-mization analysis of cycle 2 in terms of the entransy lossextremum or entropy generation minimization is unnecessary.

Bejan has pointed out that Eq. (30) denies an essential part of

the physics of the optimization process, namely, the fact that _Q H

must be free to vary [17]. Then he introduced an additional sourceof entropy generation on the outside of the visible confines of thepower plant (see Fig. 8.12 in Ref. [17]). By this method, Bejanproved that the maximum power output and the minimumentropy generation lead to the same optimal designs [17]. Thatis, the optimal operating temperatures corresponding to the twicemaximized power output can also be derived by minimizing theentropy generation rate. But note that the cases analyzed in Ref.[17] are only for n = 1 (i.e., the Chambadal’s model or C–A model).For the cases of n P 2, the proof that the maximum power outputand minimum entropy generation lead to the same design may benot as simple as that for the cases of n = 1 since we may need tointroduce several more additional sources of entropy generationon the outside of the visible confines of the n-stage power plant.More importantly, even if this proof is completed, it only meansthat the maximum power output and minimum entropy genera-tion result in the same optimal designs. But as demonstrated inSection 4, the method of maximum power output (i.e., theapproach I or II as shown in Table 2) can give only one choice ofthe intermediate temperatures corresponding to the twice maxi-mized power output. Hence, it is still impossible to find all of theoptimal intermediate operating temperatures in terms of entropygeneration minimization.

7. Concluding remarks

Entransy transfer efficiency, defined as the ratio of the entransyout to entransy in, can serve as a measure of approximation of anactual heat transfer process to a reversible one. By using theentransy transfer efficiency as an auxiliary parameter, a new andconvenient approach to determining the intermediate operatingtemperatures of n-stage endoreversible combined cycle powerplants corresponding to the twice maximized power output is pro-posed. It is found by this approach that when the temperatures ofthe hot and cold reservoirs, the total thermal conductance as wellas the stage number, n, are given, only two of these intermediatetemperatures have fixed values, while the other ones are variablewhose values can be determined from the general formula, Eq.(16). Moreover, the different choices of the variable intermediatetemperatures lead to the same amount of the twice maximized

power output. The new approach provides considerable flexibilityfor the designers in selecting the operating temperatures andappropriate working fluids with the aim of producing the twicemaximized power output.

A two-stage combined cycle power plant is taken as an exampleto illustrate the feasibility and superiority of the newly proposedapproach to the existing one. It is demonstrated that the existingtwo-step approach gives only one choice of the intermediate tem-peratures corresponding to the twice maximized power output.The newly proposed approach, by contrast, could be more efficientand convenient since it gives all the possible choices of the optimalintermediate temperatures in a step-by-step manner.

A comparison between the entransy transfer efficiency(entransy-based efficiency) and the second law efficiency(exergy-based efficiency) is performed. Both efficiencies can serveas performance parameters of heat transfer processes to measurethe approximation of an actual heat transfer process to the rever-sible one, but they have different physical meanings. Theexergy-based efficiency measures losses in availability during aheat transfer process, while the entransy-based efficiency gaugesdissipations in entransy during the process. If what we are con-cerned with is how well heat is transferred rather than how effec-tively availability is used, the entransy-based efficiency,independent of T0, may be more appropriate to evaluate the perfor-mance. Since establishing explicit equations that relate theunknown intermediate temperatures and the known parameterswith the help of the exergy-based efficiency is considerably diffi-cult, we do not take the exergy-based efficiency as the auxiliaryparameter for the determination of all the possible intermediateoperating temperatures in this study. But the entransy-based effi-ciency still has its limitation: it can only account for the imperfec-tion originating from the finite temperature difference heattransfer, and cannot gauge other imperfections that may causean actual process to deviate from the idealized one, such as frictionor unrestrained expansion during a work transfer process.

An attempt to derive the optimal intermediate temperaturesbased on the entransy loss and entropy generation analysis is alsomade. The result shows that neither the concept of entransy lossnor that of entropy generation is applicable for the determinationof all the possible intermediate operating temperatures of then-stage combined cycle power plant corresponding to the twicemaximized power output.

Conflict of interest

None declared.

Acknowledgments

This work is supported by the National Natural Science Founda-tion of China (51206079 and 51356001) and the FundamentalResearch Funds for the Central Universities (2014TS116).

Appendix A. Derivation of expressions in Table 1

In this appendix, the derivation of the expressions listed inTable 1 is given. The sketches of each model have been presentedin Table 1 and are not repeated here. Throughout the discussion,we assume the hot and cold temperature limits, TH and TL, to begiven.

A.1. Novikov’s model

For the Novikov’s model, besides the external irreversibility dueto the heat transfer process between TH and THC, the internal irre-

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 159

versibility in the process of 3-4irr was also considered. The thermalefficiency of cycle 1-2-3-4irr-1 is

g ¼ 1�_Q L

_QH

¼ 1� TLðs4irr � s1ÞTHCðs3 � s2Þ

: ðA:1Þ

According to Eq. (8), the entransy transfer efficiency of theheat transfer process between TH and THC is gentransy ¼ THC=TH.Substituting it into Eq. (A.1) yields

g ¼ 1� ð1þ iÞ TL

TH� 1gentransy

; ðA:2Þ

where i ¼ ðs4irr � s1Þ=ðs4 � s1Þ is an irreversibility parameter.

A.2. Endoreversible Carnot cycle with finite heat reservoirs

In this model, the reversible Carnot cycle is coupled to two heatreservoirs with finite heat capacities. Moreover, assume that eachof the two heat exchangers involved in this model cycle has finitethermal conductance. As a result, the outlet temperature of the hotstream, TH;out, is higher than THC, while the outlet temperature ofthe cold stream, TL;out, is lower than TLC.

Consider the heat transfer process in heat exchanger I, as shownin Fig. A1(a). The rate of entransy transfer accompanied by heattransfer from the hot reservoir (i.e., the consumed valuableresource) is

_Gin ¼Z _QH

0Thd _Q ¼

Z TH;out

TH

�CHThdTh ¼12

CH T2H � T2

H;out

� �; ðA:3Þ

where Th represents the temperature of the hot reservoir,CH ¼ _mHcp;H is its heat capacity rate.

At the same time, the rate of entransy transfer accompanied byheat transfer into the working fluid (i.e., the produced valuableresource) is

_Gout ¼Z _QH

0Tcd _Q ¼ Tc

Z _QH

0d _Q ¼ THC

_QH; ðA:4Þ

where Tc represents the temperature of the working fluid and isalways equal to THC during the heat transfer process.

In addition, considering linear heat transfer law, the heat trans-fer rate, _QH, in Eq. (A.4), can be given by

_Q H ¼ CH TH � TH;outð Þ: ðA:5Þ

Substituting Eqs. (A.3) and (A.4) into gentransy ¼_Gout_Gin

and using Eq.

(A.5), the entransy transfer efficiency of heat transfer process I is

Fig. A1. Sketch of two heat exchangers: (a) heat transfer process occurs betweenthe hot reservoir and the working fluid; (b) heat transfer process occurs betweenthe working fluid and the cold reservoir.

gentransy;I ¼2THC

TH þ TH;out: ðA:6Þ

Similarly, the entransy transfer efficiency of the heat transferprocess between the working fluid and the cold reservoir(Fig. A1(b)) can be obtained as

gentransy;II ¼TL þ TL;out

2TL: ðA:7Þ

The heat transfer rate, _QH, can also be calculated by the LMTD(logarithmic mean temperature difference), that is,

_QH ¼ ðUAÞHTH � TH;out

ln TH�THCTH;out�THC

; ðA:8Þ

where ðUAÞH is the thermal conductance of heat exchanger I.To find the outlet temperature of the hot stream, TH;out, combin-

ing Eqs. (A.5) and (A.8) gives

TH;out ¼ THC þ TH � THCð Þe�ðUAÞH

CH : ðA:9Þ

The outlet temperature of the cold stream, TL;out, can be derivedin a same manner, that is,

TL;out ¼ TLC � TLC � TLð Þe�ðUAÞL

CL ; ðA:10Þ

where ðUAÞL is the thermal conductance of heat exchanger II, and CL

is the heat capacity rate of the cold reservoir.Substituting Eqs. (A.9) and (A.10) into (A.6) and (A.7), respec-

tively, the two entransy transfer efficiencies become

gentransy;I ¼2THC

TH þ TH;out¼ 2

TH

THCþ 1þ TH

THC� 1

� �e�

ðUAÞHCH

� �1

ðA:11Þ

and

gentransy;II ¼TL þ TL;out

2TL¼ 1

2TL

TLCþ 1þ TL

TLC� 1

� �e�

ðUAÞLCL

� : ðA:12Þ

Based on the above two equations, the ratios, THTHC

and TLTLC

can be

obtained as

TH

THC¼ 2

gentransy;If H� gH ðA:13Þ

and

TL

TLC¼

2gentransy;II

f L� gL

� ��1

; ðA:14Þ

where f H ¼ 1þ e�ðUAÞH=CH , f L ¼ 1þ e�ðUAÞL=CL , gH ¼ 1� e�ðUAÞH=CH �

=

1þ e�ðUAÞH=CH �

and gL ¼ 1� e�ðUAÞL=CL �

= 1þ e�ðUAÞL=CL �

, which arefixed for given ðUAÞH, ðUAÞL, CH and CL.

Hence the thermal efficiency of the endoreversible Carnot cyclewith finite heat reservoirs, g ¼ 1� TLC=THC, can be rewritten as

g ¼ 1� TL

TH

2gentransy;If H

� gH

!�

2gentransy;II

f L� gL

� ��1

: ðA:15Þ

A.3. Endoreversible Brayton cycle with finite heat reservoirs

The Brayton cycle is an ideal cycle that describes the operationof a gas turbine engine. It consists of two constant pressure andtwo isentropic processes, as shown in Table 3 (1-2-3-4-1). Theworking fluid absorbs heat from the hot reservoir and releases heatto the cold reservoir through two counter flow heat exchangerswith the numbers of heat transfer units NTUH and NTUL, respec-tively. The heat capacity rates of the hot and cold reservoirs as wellas the working fluid are respectively CH, CL and Cm, and assume

160 J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161

that they are given. The inlet and outlet temperatures of the hotreservoir are TH and TH;out, while for cold one, they are TL and TL;out.

As shown in Eq. (A.3), the entransy flow out of the hot reservoirduring the heat transfer process between the hot reservoir and theworking fluid is

_Gin ¼12

CHT2H �

12

CHT2H;out: ðA:16Þ

At the same time, the entransy flow into the working fluid is

_Gout ¼12

CmT23 �

12

CmT22: ðA:17Þ

The energy conservation law gives

_Q H ¼ CHðTH � TH;outÞ ¼ CmðT3 � T2Þ: ðA:18Þ

Accordingly, the entransy transfer efficiency of this heat trans-fer process is

gentransy;I ¼_Gout

_Gin

¼ T2 þ T3

TH þ TH;out: ðA:19Þ

In the same way, the entransy transfer efficiency of the heattransfer process between the working fluid and the cold reservoiris

gentransy;II ¼TL þ TL;out

T1 þ T4: ðA:20Þ

For the Brayton cycle, there should be T2=T1 ¼ T3=T4. Thus, itsthermal efficiency, g ¼ 1� T1=T2, can be rewritten as

g ¼ 1� T1 þ T4

T2 þ T3: ðA:21Þ

From Eqs. (A.19)–(A.21), we obtain

g ¼ 1� 1gentransy;I � gentransy;II

� TL þ TL;out

TH þ TH;out

� �: ðA:22Þ

The heat transfer rate between the hot reservoir and the work-ing fluid can also be expressed as

_Q H ¼ CH;mineH TH � T2ð Þ; ðA:23Þ

where CH;min ¼minðCH;CmÞ, eH is the effectiveness with theexpression

eH ¼1� exp �NTUH 1� CH�

� �h i1� CH� exp �NTUH 1� CH�

� �h i : ðA:24Þ

In Eq. (A.24), CH� ¼minðCH;CmÞ=maxðCH; CmÞ and NTUH ¼ðUAÞH=minðCH;CmÞ.

Combining Eqs. (A.18), (A.19) and (A.23) to eliminate T2, T3 andTH;out, the heat transfer rate, _QH, is determined to be

_Q H ¼2TH 1� gentransy;I

� �1

Ca� gentransy;I

CH

; ðA:25Þ

where

1Ca¼ 2

CH;mineH� 1

Cm: ðA:26Þ

In the similar lines, the heat transfer rate between the workingfluid and the cold stream can be expressed as

_Q L ¼ CLðTL;out � TLÞ ¼ CmðT4 � T1Þ ¼ CL;mineLðT4 � TLÞ; ðA:27Þ

where CL;min ¼minðCL;CmÞ, eL is the effectiveness with theexpression

eL ¼1� exp �NTUL 1� CL�

� �h i1� CL� exp �NTUL 1� CL�

� �h i : ðA:28Þ

In Eq. (A.28), CL� ¼minðCL;CmÞ=maxðCL;CmÞ and NTUL ¼ ðUAÞL=minðCL;CmÞ. Combining Eqs. (A.27) and (A.20) to eliminate T1, T4

and TL;out leads to

_Q L ¼2TL 1� gentransy;II

� �gentransy;II

Cb� 1

CL

; ðA:29Þ

where1Cb¼ 2

CL;mineL� 1

Cm: ðA:30Þ

Substituting Eqs. (A.18) and (A.27) into (A.22) to eliminate TH;out

and TL;out, yields

g ¼ 1� 1gentransy;I � gentransy;II

� 2TL þ _Q L=CL

2TH � _Q H=CH

!: ðA:31Þ

Then, using Eqs. (A.25) and (A.29) to eliminate _QH and _Q L in Eq.(A.31), we finally get

g ¼ 1� TL

TH�

CLCb� 1

CHCa� 1

!�

CHCa

1gentransy;I

� 1CLCb

gentransy;II � 1

0@

1A: ðA:32Þ

where the parameters, CH, CL, Ca, Cb are given values in many cases.

A.4. Endoreversible Stirling cycle with finite heat reservoirs

The Stirling cycle is another thermodynamic cycle which wasfirst patented in 1816 by Robert Stirling. It consists of two isother-mal processes 1-2 and 3-4 and two constant-volume processes 2-3and 4-1. Two externally irreversible heat transfer processes occurbetween the working fluid and two reservoirs with inlet tempera-tures of TH and TL, respectively, through two counter flow heatexchangers, as shown in Table 3. The heat capacity rates of thehot and cold reservoirs are CH and CL, and assume that they aregiven. The inlet and outlet temperatures of the hot reservoir areTH and TH;out, while for cold one, they are TL and TL;out.

In the two constant-volume processes, heat is transferredbetween the working fluid and a regenerator. For an ideal regener-ation, the heat stored in the regenerator during the process 4-1 isentirely released to the working fluid during the process 2-3. Nev-ertheless, in the real conditions, there is a regenerative heat loss,D _QR, during the two regenerative processes, which can beexpressed as [30,36]

D _QR ¼ CR 1� eRð Þ THS � TLSð Þ; ðA:33Þ

where CR is the heat capacity rate of the working fluid during theregenerative processes, eR is the effectiveness of the regenerator,THS and TLS are the temperatures of the working fluid in the process3-4 and 1-2, respectively. Therefore, besides the irreversibility orig-inating from the heat transfer processes in the two heat exchangers,we should also take into account the irreversibility due to the finiteheat transfer in the regenerative processes where the amount ofheat loss can be regarded to transfer from THS to TLS directly, whoseentransy transfer efficiency is

gentransy;R ¼D _Q RTLS

D _Q RTHS

¼ TLS

THS: ðA:34Þ

Moreover, taking the heat loss into account, the net heatabsorbed from the hot reservoir and that released to the cold reser-voir per unit time are

_QH ¼ _Q 3�4 þ D _Q R ðA:35Þ

J. Wu / International Journal of Heat and Mass Transfer 91 (2015) 150–161 161

and_Q L ¼ _Q 1�2 þ D _QR; ðA:36Þ

where _Q3�4 ¼ _mRgTHS lnðVmax=VminÞ and _Q1�2 ¼ _mRgTLS lnðVmax=

VminÞ, _m and Rg are respectively the mass flow rate and gas constantof the working fluid, Vmax and Vmin are respectively the maximumand minimum volumes of the working fluid during the constantvolume processes.

Based on Eqs. (A.34)–(A.36), the thermal efficiency of the Stir-ling cycle is given by

g ¼_W

_Q H

¼_Q 3�4 � _Q 1�2

_Q 3�4 þ D _Q R

¼ 1� _Q 1�2= _Q 3�4

1þ 1� gentransy;R

� �CR 1� eRð Þ= _mRg ln Vmax=Vminð Þ

� ; ðA:37Þ

where 1� _Q1�2= _Q3�4 is the thermal efficiency of the Stirling enginewithout the regenerative heat loss. The derivation of Eq. (A.15) canbe repeated here to obtain

g ¼1� TL

TH

2gentransy;I f H

� gH

h i� 2gentransy;II

f L� gL

h i�1

1þ 1�gentransy;Rð ÞCR 1�eRð Þ_mRg ln Vmax=Vminð Þ

; ðA:38Þ

where gentransy;I, gentransy;II are the entransy transfer efficiencies of thetwo heat transfer processes between the working fluid and the heatreservoirs, f H, f L, gH and gL are the same as in Eq. (A.15). In partic-ular, if the regeneration is ideal, we have eR ¼ 1, and then Eq.(A.38) reduces to Eq. (A.15).

A.5. Endoreversible Carnot refrigeration or heat pump cycles

In this model, the temperatures of the heat reservoirs, TH and TL,are assumed to be unchanged during the heat transfer processeswith the working fluid (refrigerant).

The COP of the refrigeration cycle is

COP ¼_Q L

_W¼

_Q L

_QH � _Q L

¼ 1THCTLC� 1

; ðA:39Þ

where _Q L is the heat transfer rate from the cold reservoir to the

working fluid which is usually referred to as the cooling load, _QH

is the heat transfer rate from the working fluid to the hot reservoir,_W is the power input, THC and TLC are respectively the temperatures

of the working fluid during the two isothermal processes of thereversed Carnot cycle.

Substituting the entransy transfer efficiencies of the two heattransfer processes, gentransy;I ¼ TH=THC and gentransy;II ¼ TLC=TL intoEq. (A.39) yields

COP ¼ 11

gentransy;I� 1gentransy;II

� THTL� 1

: ðA:40Þ

If the reversed Carnot cycle is used as a heat pump, its COPbecomes

COP ¼_Q H

_W¼

_Q H

_Q H � _Q L

¼ 11� gentransy;I � gentransy;II �

TLTH

: ðA:41Þ

References

[1] P. Chambadal, Les Centrales Nucleaires, Armand Colin, Paris, 1957.[2] I.I. Novikov, The efficiency of atomic power stations, J. Nucl. Energy 7 (1958)

125–128.

[3] F.L. Curzon, B. Ahlborn, Efficiency of a Carnot engine at maximum poweroutput, Am. J. Phys. 43 (1975) 22–24.

[4] B. Andresen, R.S. Berry, A. Nitzan, P. Salamon, Thermodynamics in finite time. I.The step-Carnot cycle, Phys. Rev. A 15 (1977) 2086–2093.

[5] P. Salamon, B. Andresen, R.S. Berry, Thermodynamics in finite time. II.Potentials for finite-time processes, Phys. Rev. A 15 (1977) 2094–2102.

[6] S. Sieniutycz, P. Salamon, Finite-time Thermodynamics and Thermoeconomics,Taylor & Francis, New York, 1990.

[7] A. Bejan, Entropy Generation Minimization: The Method of ThermodynamicOptimization of Finite-Size Systems and Finite-time Processes, CRC Press, NewYork, 1995.

[8] A. Durmayaz, O.S. Sogut, B. Sahin, H. Yavuz, Optimization of thermal systemsbased on finite-time thermodynamics and thermoeconomics, Prog. EnergyCombust. Sci. 30 (2004) 175–217.

[9] L.G. Chen, F.R. Sun, Advances in Finite Time Thermodynamics: Analysis andOptimization, Nova Publishers, New York, 2004.

[10] S. Bandyopadhyay, N.C. Bera, S. Bhattacharyya, Thermoeconomic optimizationof combined cycle power plants, Energy Convers. Manage. 42 (2001) 359–371.

[11] J.C. Chen, The general performance characteristics of an n-stage combinedrefrigeration system affected by multi-irreversibilities, J. Phys. D Appl. Phys. 32(1999) 1462–1468.

[12] X.C. Xuan, Optimum staging of multistage exo-reversible refrigerationsystems, Cryogenics 43 (2003) 117–124.

[13] H. Sahraie, M.R. Mirani, M.H. Ahmadi, M. Ashouri, Thermo-economic andthermodynamic analysis and optimization of a two-stage irreversible heatpump, Energy Convers. Manage. 99 (2015) 81–91.

[14] M.H. Rubin, B. Andresen, Optimal staging of endoreversible heat engines, J.Appl. Phys. 53 (1) (1982) 1–7.

[15] Govind Maheshwari, S. Chaudhary, S.K. Somani, Performance analysis ofendoreversible combined Carnot cycles based on new maximum efficientpower (MEP) approach, Int. J. Low-Carbon Technol. 5 (2010) 1–6.

[16] A. Bejan, Theory of heat transfer-irreversible power plants–II. The optimalallocation of heat exchange equipment, Int. J. Heat Mass Transfer 38 (1995)433–444.

[17] A. Bejan, Entropy Generation Minimization, CRC Press, Florida, 1996. 209, 227–231.

[18] A. Bejan, Thermodynamic optimization alternatives: minimization of physicalsize subject to fixed power, Int. J. Energy Res. 23 (1999) 1111–1121.

[19] Z.Y. Guo, H.Y. Zhu, X.G. Liang, Entransy-A physical quantity describing heattransfer ability, Int. J. Heat Mass Transfer 50 (2007) 2545–2556.

[20] Q. Chen, X.G. Liang, Z.Y. Guo, Entransy theory for the optimization of heattransfer–a review and update, Int. J. Heat Mass Transfer 63 (2013) 65–81.

[21] Z.Y. Guo, X.G. Cheng, Z.Z. Xia, Least dissipation principle of heat transportpotential capacity and its application in heat conduction optimization, Chin.Sci. Bull. 48 (4) (2003) 406–410.

[22] X.T. Cheng, X.G. Liang, Entransy loss in thermodynamic processes and itsapplication, Energy 44 (2012) 964–972.

[23] X.T. Cheng, Q. Chen, G.J. Hu, X.G. Liang, Entransy balance for the closed systemundergoing thermodynamic processes, Int. J. Heat Mass Transfer 60 (2013)180–187.

[24] X.T. Cheng, X.G. Liang, Entransy variation associated with work, Int. J. HeatMass Transfer 81 (2015) 167–170.

[25] W.H. Wang, X.T. Cheng, X.G. Liang, Entropy and entransy analyses andoptimization of the Rankine cycle, Energy Convers. Manage. 68 (2013) 82–88.

[26] A.B. Yang, L.G. Chen, S.J. Xia, F.R. Sun, The optimal configuration ofreciprocating engine based on maximum entransy loss, Chin. Sci. Bull. 59(2014) 2031–2038.

[27] Y. Zhu, Z. Hu, Y. Zhou, L. Jiang, L. Yu, Applicability of entropy, entransy andexergy analyses to the optimization of the organic Rankine cycle, EnergyConvers. Manage. 88 (2014) 267–276.

[28] T. Li, W. Fu, J. Zhu, An integrated optimization for organic Rankine cycle basedon entransy theory and thermodynamics, Energy 72 (2014) 561–573.

[29] K.H. Kim, K. Kim, Comparative analyses of energy–exergy–entransy for theoptimization of heat-work conversion in power generation systems, Int. J. HeatMass Transfer 84 (2015) 80–90.

[30] I. Tlili, Finite time thermodynamic evaluation of endoreversible Stirling heatengine at maximum power conditions, Renew. Sustainable Energy Rev. 16(2012) 2234–2241.

[31] L.G. Chen, J.L. Zheng, F.R. Sun, C. Wu, Performance comparison of anendoreversible closed variable temperatures heat reservoir Brayton cycleunder maximum power density and maximum power conditions, EnergyConvers. Manage. 43 (2002) 33–43.

[32] M.J. Moran, Availability Analysis: A Guide to Efficient Energy Use, Prentice-Hall, Englewood Cliffs, New Jersey, 1989. pp. 86.

[33] Y.A. Cengel, A.M. Boles, Thermodynamics: An Engineering Approach, fifth ed.,McGraw-Hill, Boston, 2002. pp. 424.

[34] K. Wark Jr., D.E. Richards, Thermodynamics, sixth ed., McGraw-Hill, Boston,1999. pp. 512.

[35] X.T. Cheng, X.G. Liang, Optimization of combined endoreversible Carnot heatengines with different objectives, Chin. Phys. B 24 (2015) 060510.

[36] S.C. Kaushik, S. Kumar, Finite time thermodynamic analysis of endoreversibleStirling heat engine with regenerative losses, Energy 25 (2000) 989–1003.