App Energy 2004 FTT Lect

APPLIED

www.elsevier.com/locate/apenergy

Applied Energy 79 (2004) 27–40ENERGY

Finite-time heat-transfer analysis andgeneralized power-optimization

of an endoreversible Rankine heat-engine

Abdul Khaliq *

Department of Mechanical Engineering, Faculty of Engineering and Technology,

Jamia Millia Islamia, New Delhi 110025, India

Accepted 4 December 2003

Available online 5 February 2004

Abstract

This paper reports the results of a study carried out for the power optimization of a

Rankine-cycle heat-engine using finite-time thermodynamic theory. This study extends the

recent flurry of publications in heat-engine efficiency under the maximum power condition by

incorporating the optima of heat conductance and heat capacitance ratios. While maximizing

the instantaneous power output, it is shown that there is an optimum balance between the sizes

of heat exchangers, between the heat capacity rates of heating and cooling fluid as well as

temperature differences between the engine and thermal reservoirs. The results indicate that

power output increases significantly with the increase in heat capacity rate of the heating fluid,

but the thermal efficiency at maximum power remains constant. The effects of thermal con-

ductance of the hot-side heat-exchanger on the power output and thermal efficiency are in-

significant. The theoretical efficiency formulated in this study is much closer to that actually

observed in well-designed power plants.

� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

Classical thermodynamics typically considers equilibrium states and limits on

processes between equilibrium states. Time is not considered in the formulation, andreversible limits are treated, for which the rates are infinitely slow. The reversible

* Tel.: +91-11-26328717.

E-mail address: [email protected] (A. Khaliq).

0306-2619/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apenergy.2003.12.003

mail to: [email protected]

Nomenclature

a external thermal-conductance allocation ratio

b external thermal-capacitance ratio

AH surface area of heat exchanger operating between heat source and heatengine (m2)

AL surface area of heat exchanger operating between heat sink and heat

engine (m2)

CPH specific heat of heat-source fluid (kJ/kgK)

CPL specific heat of heat-sink fluid (kJ/kgK)

HE,LE constants defined in text

LMTDH log mean temperature-difference between heat source and heat engine

at heat exchanger_mH mass flow rate of heat source at heat exchanger (kg/s)_mL mass flow rate of heat sink at heat exchanger (kg/s)

P power generated from heat engine (kW)

Pmax optimum power generated by finite-time heat engine (kW)

QH heat transferred from heat source to heat engine (kJ)

QL heat transferred from heat engine to heat sink (kJ)_QH rate of input heat transfer (kW)_QL rate of output heat transfer (kW)r ðt12 þ t34Þ=ðtH þ tcÞS entropy (kJ/K)

T temperature (K)

Ta entropic average temperature (K)

Tc entropic average temperature of working substance in condenser (K)

TH1 inlet temperature of heating fluid at heat exchanger with

finite-heat-capacity rate (K)

TH2 outlet temperature of heating fluid at heat exchanger withfinite-heat-capacity rate (K)

TL1 inlet temperature of cooling fluid at heat exchanger with


TL2 outlet temperature of cooling fluid at heat exchanger with


Tw entropic average temperature of working substance in evaporator (K)

tc time required to transfer QL (s)

tcy total time required for whole heat-engine cycle (s)tH time required to transfer QH (s)

t12 time required for isentropic-expansion process (s)

t34 time required for isentropic-pumping process (s)

UH overall heat-transfer co-efficient at heat exchanger between heat source

and heat engine (kW/m2 K)

UL overall heat-transfer co-efficient at heat exchanger between heat sink

and heat engine (kW/m2 K)

28 A. Khaliq / Applied Energy 79 (2004) 27–40

W output power generated by heat engine (kW)

x TH1 � Twy Tc � TL1gCarnot classical Carnot-efficiencygopt thermodynamic-cycle efficiency at maximum power

Definitions

aUHAH

_mLCPH

bULAL

_mLCPL

hð1� aÞCe

ð1� bÞ _mCp

r b _mCp

s ð1� bÞ _mCp

/aCe

b _mCp

A. Khaliq / Applied Energy 79 (2004) 27–40 29

Carnot cycle is generally recognized as the best cycle operating between two thermal

reservoirs. However, the use of reversible processes as standards of performance in

industry is not desirable because a reversible process must be carried out at an in-

finitely slow pace. Since power produced by a heat engine is work divided by time, a

finite amount of work produced by the engine over an infinite time delivers no

power. No practising engineer wants to design an engine that runs infinitely slowly

without producing power. So, the need to produce power in real energy-conversion

devices is the first reason and to consider heat transfer interaction with the sur-rounding is the second reason for the evolution of finite-time thermodynamics.

Denton [14] concluded that a major objective of finite-time thermodynamics is to

understand irreversible finite-time heat-transfer processes and to establish bounds on

efficiency and maximum work for such processes. Another primary goal of finite-

time thermodynamics is to establish general operating principles for a system which

serves as model for real processes.

Finite-time thermodynamics appears to have its origin in two independently

prepared published papers, one by Novikov [1] and the other by Chambadal [2].Another, apparently independently prepared, paper was published by Curzon and

Ahlborn [3] on the efficiency of the Carnot-engine at maximum power. They con-

sidered the case of finite rates of heat transfer to and from a Carnot-engine. After

maximizing the power output, they derived a simple expression for the efficiency that

was different from the well known Carnot efficiency. Since finite-time thermody-

namics was introduced, the effect of external irreversibility due to finite-time heat

transfer on the performance of various cycles has been studied by several authors.

Bejan [4,5] reported the early history of finite-time thermodynamics in combinationwith the history of entropy generation minimization. Onderchen et al. [6] analysed

the maximum work from a finite reservoir by using sequential Carnot heat-engines.


Wu [7] studied a finite-time heat-engine, with both the heat source and heat sink

having finite heat-capacity rates and obtained numerical solutions for the efficiency

at maximum power. Lee et al. [8] analysed the finite-time Carnot heat-engine op-

erating between two reservoirs with finite heat-capacity rates for maximum power

and efficiency at maximum power. Bejan [9,10] dealt with the power optimization of

continuous endoreversible Carnot cycles and showed the effect of the allocation ofhot and cold end thermal reservoir heat conductances on the power output of a

continuous heat-engine operating between infinite thermal reservoirs, and also de-

veloped models for power plants that generate minimum entropy while operating at

maximum power. Since the Carnot heat-engine itself does not have any real world

counterpart, Wu [11], Blank et al. [12] and, Blank and Wu [13] went on to describe

the finite-time thermodynamic optimizations of Rankine, Stirling and Ericsson heat

engines, which do have practical counter parts. But their study was limited to heat

engines with both the heat source and heat sink having infinite heat-capacity rates,and obtained only the optimal temperature of heating and cooling fluids for maxi-

mum power.

In this paper, the optimization work performed by Wu [11] and others [14] for the

infinite heat-capacity rates model is extended to a finite heat-capacity rate Rankine

heat-engine by following a log mean temperature-difference formulation. The power

is maximized with heating and cooling fluid temperatures, heat conductance allo-

cation ratio, and heat capacitance ratio, for the most general optimization of a

Rankine heat-engine. Some conclusions, that have more realistic significance thanthose of previous studies on this subject, have been obtained.

2. System description

A reversible Rankine heat-engine is connected to heating and cooling fluids with

finite heat-capacity rates. The finite-time heat transfer processes are irreversible. This

TH1

Tw5

TH2

1

23

4

TL1

TL2

Entropy (s)

Tem

pera

ture

(T

)

Tc

Fig. 1. Temperature–entropy diagram of a simple Rankine heat-engine.


cycle is shown in Fig. 1 on the temperature–entropy plane. A finite mass of heating

fluid acts as a source of energy. Heat is transferred from the heating fluid to the

Rankine heat-engine which converts a fraction of the heat to work and rejects the

rest to a heat sink, which is a finite mass of cooling fluid. Because both the source

and the sink of the energy have finite masses, their temperatures change as a result of

the heat-transfer processes. The temperature of the source is initially TH1 and cools toTH2 as a result of giving up heat QH. The temperature of the sink is initially TL1 andwarms up to TL2 as a result of receiving QL.

3. Finite-time thermodynamic optimization

Using an entropic average temperature, the ideal Rankine cycle can be modified

to a Carnot cycle to achieve the theoretical formulae. Since the area under theprocess 4–5–1 in the T–s diagram of Fig. 1 represents the amount of heat added to

the Rankine cycle, we can make this area equal to the area under the horizontal line

with an entropic average temperature of heat addition.

The entropic average temperature Ta can be defined as follows

Ta ¼DQDS

; ð1Þ

where Q and S are the heat and entropy, respectively. From Eq. (1), the entropic

average temperature of heat addition, Tw becomes

Tw ¼ H1 � H4

S1 � S4; ð2Þ

where H denotes the enthalpy, when the working fluid enters the expander as dry

saturated vapour and leaves it as superheated vapour; the entropic average tem-

perature of the condensing working-fluid can also be represented in the same

manner. The modified Rankine engine becomes a Carnot-engine operating between

Tw and Tc as shown in Fig. 2.The heat engine operates in a cyclic mode with fixed time tcy allotted for each

cycle. Thus, after time tcy has elapsed, the working fluid returns to its initial state.

The modified Rankine heat-engine consists of two isothermal (4–1, 2–3) and two

isentropic processes (1–2, 3–4).

We assume that the inlet temperatures of a heat source and a heat sink (TH1, TL1)are fixed, but heat conductances (UHAH, ULAL) and heat capacitances

( _mHCPH; _mLCPL) are changing. Therefore, the temperature distributions of the

heating and cooling fluids are not constant through the heat exchangers. The rate ofheat flow from the heat source to the heat engine is proportional to the log mean

temperature-difference (LMTDH) and is equivalent to the decreasing rate of heat

input from the heating fluid.

Tw4

TH2

1

Entropy (s)

Tem

pera

ture

(T

)

3 2

TL1

TL2

TH1

Tc

Fig. 2. Temperature–entropy diagram of a modified Rankine heat-engine.


If the heat input lasts tH per cycle, the rate of heat input is

_QH ¼ QH

tH¼ UHAHLMTDH ¼ _mHCPHðTH1 � TH2Þ; ð3Þ

where LMTDH ¼ ½ðTH1 � TwÞ � ðTH2 � TwÞ�= ln ðTH1�TwÞðTH2�TwÞ

h i.

Similarly, for condensation, the rate of heat flow from the heat engine to the heat

sink is

_QL ¼ QL

tc¼ ULALLMTDL ¼ _mLCPLðTL2 � TL1Þ; ð4Þ

where LMTDL ¼ ½ðTc � TL1Þ � ðTc � TL2Þ�= ln½ðTc � TL1Þ=ðTc � TL2Þ�The time taken to complete the whole cycle is

tcy ¼ tH þ tc þ t12 þ t34 ¼ ðtH þ tcÞð1þ rÞ: ð5Þ

Since the times t12 and t34 required for the two isentropic processes of the cycle arenegligibly small relative to tH and tc, Eq. (5) reduces to

tcy ¼ tH þ tc: ð6Þ

Using Eqs. (3) and (4), the total time tcy is given by the expression

tcy ¼QH

_mHCPHðTH1 � TH2Þþ QL

_mLCPLðTL2 � TL1Þ: ð7Þ

Using Eq. (3), the outlet temperature of the heating fluid is given as a function of

Tw and TH1 as


TH2 ¼ Tw þ ðTH1 � TwÞe�a: ð8Þ
Similarly using (4),
TL2 ¼ Tc � ðTc � TL1Þe�b: ð9Þ
Using Eqs. (8) and (9) to eliminate TH2 and TL2 in Eq. (7), the total cycle time
becomes

tcy ¼QH

HEðTH1 � TwÞþ QL

LEðTc � TL1Þ; ð10Þ

where

HE ¼ _mHCPHð1� e�aÞ;

LE ¼ _mLCPLð1� e�bÞ:
The net work output W is given by
W ¼ QH � QL: ð11Þ
Since the modified Rankine cycle is endoreversible and operates between the
temperatures Tw and Tc, the net entropy change is zero and

QH

Tw¼ QL

Tc: ð12Þ

From Eqs. (11) and (12)

QH ¼ WTw=ðTw � TcÞ;QL ¼ WTc=ðTw � TcÞ:

ð13Þ

Using Eqs. (13) for QH and QL, Eq. (10) becomes

tcy ¼WTw

HEðTH1 � TwÞðTw � TcÞ

�þ WTc

LEðTc � TL1ÞðTw � TcÞ

��1

; ð14Þ

and the average power is given by

P ¼ Wtcy

¼ TwHEðTH1 � TwÞðTw � TcÞ

�þ TcLEðTc � TL1ÞðTw � TcÞ

�: ð15Þ

Eq. (15) can also be written as

P ¼ HELExyðTH1 � TL1 � x� yÞLETH1y þHETL1xþ xyðHE� LEÞ ; ð16Þ

where x ¼ TH1 � Tw and y ¼ Tc � TL1.Assuming the total conductance and total capacitance inventory of the heat

engine to be Ce and _mCP respectively, such that

UHAH þ ULAL ¼ Ce; ð17Þ

_mHCPH þ _mLCPL ¼ _mCP: ð18Þ


Eqs. (17) and (18) give

UHAH ¼ aCe; ð19Þ

ULAL ¼ ð1� aÞCe; ð20Þ

_mHCPH ¼ b _mCP; ð21Þ

_mLCPL ¼ ð1� bÞ _mCP; ð22Þ

where a and b are the heat-conductance allocation ratio and heat-capacitance ratio

respectively.

Substituting Eqs. (19)–(22) into Eq. (16), then using the terms defined earlier, oneobtains an alternate expression for the power output of a heat engine as

P ¼ ð1� e�/Þsð1� e�hÞxyðTH1 � TL1 � x� yÞsð1� e�hÞTH1y þ rð1� e�/ÞTL1xþ xyrð1� e�/Þ � sð1� e�hÞ : ð23Þ

In the present model, we assume that the inlet temperatures of the heating and

cooling fluids are fixed, but the heat-capacity rates and heat conductances of the heat

sink and heat source are varying, and we neglect all dissipative processes, such as

friction and turbulence.

For fixed values of TH1 and TL1, this expression for power indicates that the power

output is a function of x, y, a and b and the power can be maximized in two ways.

For given x and y, the expression for power in Eq. (23) is a function of �a� and �b�only.

Power from this finite-time Rankine heat-engine can in principle be maximized

after using Eq. (23) by requiring

oPoa

¼ 0 andoPob

¼ 0; ð24Þ

and solving for the optimal values of �a� and �b�. The result is

a ¼ b ¼ 1

2: ð25Þ

After substituting these optimal values of a and b into Eq. (23), the corresponding

maximum power is

Pmax ¼12_mCP 1� Ce

_mCp

� �xyðTH1 � TL1 � x� yÞ

TH1y þ TL1x: ð26Þ

Another aspect that is brought to light by the model considered in this study is

that there exist optimal temperatures of the heating and cooling fluids along with theoptimal heat conductance and heat capacitance ratios, such that the power output is

maximized once more.

Setting the derivatives


oPox

¼ 0 andoPoy

¼ 0; ð27Þ

yields the optimal temperatures of the heating and cooling fluids as

Tw ¼ TH1 �TH1

21

��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTL1=TH1

p �; ð28Þ

Tc ¼ TL1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiTL1TH1

p

21

��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTL1=TH1

p �: ð29Þ

After substituting these optimal values into Eq. (26), the twice maximized power

output may be obtained as

Pmax;max ¼ _mHCPH 1

�� e

� UHAH_mHCPH

� ffiffiffiffiffiffiffiTH1

p�

ffiffiffiffiffiffiffiTL1

p

2

� �2: ð30Þ

In conclusion, in order to operate at maximum power, there must be not only a

balance between the temperatures of the heating and cooling fluids ðTwÞopt and

ðTcÞopt, but also a balance between the heat conductance ratio aopt (the sizes of thehot and cold-end heat-exchangers) and the heat-capacitance ratio bopt (the mass flow

rate of the hot and cold end thermal reservoirs).

The thermal efficiency of the heat engine is given by

g ¼ 1� TcTw

: ð31Þ

Using Eqs. (28) and (29) in Eq. (31), the efficiency at twice the maximum power

(optimal efficiency) may be obtained as

gopt ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTL1=TH1

p: ð32Þ

The most interesting feature of the above result is that the optimal operating

temperatures and the efficiency at maximum power depend only on the initial tem-

peratures of the heating and cooling fluids, but are independent of the heat capacity

rates and the heat conductances of the heat source/sink. But the maximum power

obtained in expression (30) depends on the heat conductance and heat capacity rate

of the heat source as well as on the initial temperatures of the heating and coolingfluids. These formulae provide the very simple basis for the preliminary design of a

Rankine heat-engine.

These relations acount for the heat-engine irreversibilities associated with the

finite temperature-difference between the working fluids and source/sink reservoirs.

Other sources of irreversibilities are the non-isentropic expansion and compression

of the working fluids in the turbine and the pump. Since pump work can be neglected

compared to the turbine output, the isentropic efficiency of the turbine becomes the

most important factor of those considered affecting the overall efficiency of theRankine cycle engines employing low-grade energy. Therefore, in order to afford a

much more accurate estimation of the efficiency of real heat engines, the following

relation can be used

Table

The ob

Seri

1

2

3

4

5

6

7

Table

Effect

UHA

TH1

(K)

358

400

523

643

663

740

783

Table

Effect

UHA

TL1(K)

298

303

308

313

323

340

353


gopt;act ¼ goptgT;

where gT represents the turbine efficiency.

The efficiency at maximum power formulated in this study is compared with

observed efficiencies of actual power plants in Tables 1–6 and Figs. (3 and 4) which

1

served efficiencies of seven power-plants [4,8,11]

al no. Power plant TH1

(K)

TL1(K)

gopt(%)

gobs(%)

gCarnot(%)

Dead Sea solar thermal-power plant 358 300 8.45 8.06 16.19

Ein Boqeq solar thermal-power plant 400 303 12.96 11.90 24.25

Landerello (Italy), geothermal steam power plant 523 353 17.84 16.0 32.50

West Thurrock UK, 1962 Conventional coal fired

steam plant

643 298 31.92 28.0 53.65

Central steam power station in the UK, 1936 663 298 32.95 29.65 55.05

Steam power plant in the USA 1956 740 298 36.54 32.4 59.73

Combined cycle (steam and mercury) plant in the

USA, 1949

783 298 38.3 34.3 61.9

2

of TH1

H ¼ 1 kW/K _mHCPH ¼ 0:2 kW/K TL1 ¼ 300 K

Tw(K)

Tc(K)

TH2

(K)

TL2(K)

QH

(kW)

QL

(kW)

Pmax

(kW)

gopt(%)

gobserved(%)

342.86 313.86 342.96 313.76 3.008 2.753 0.127 8.45 8.06

373.20 323.20 373.38 323.04 5.324 4.608 0.356 13.39 12.39

459.50 348.05 459.92 347.72 12.616 9.544 1.529 22.42 23.20

541.10 369.60 541.78 369.13 20.240 13.826 3.207 31.69 30.90

554.50 373.00 555.23 372.50 21.550 14.500 3.527 32.73 31.40

605.58 385.58 606.48 385.00 26.704 17.000 4.850 36.32 35.80

633.80 392.33 634.8 391.70 29.640 18.340 5.646 38.10 37.70

3

of TL1

H ¼ 1 kW/K _mHCPH ¼ 0:2 kW/K TH1 ¼ 523 K

Tw(K)

Tc(K)

QH

(kW)

QL

(kW)

TH2

(kW)

TL2(kW)

Pmax

(kW)

gopt(%)

458.90 346.40 12.734 9.614 459.33 346.07 1.5605 24.50

460.54 350.54 12.408 9.444 460.96 350.22 1.4812 23.88

462.17 354.62 12.084 9.270 462.58 354.35 1.4047 23.25

463.79 358.80 11.762 9.098 464.18 358.49 1.3307 22.63

467.00 367.00 11.124 8.740 467.37 366.70 1.1905 21.41

472.34 380.84 10.060 8.112 472.68 380.56 0.9740 19.37

476.33 391.33 9.272 7.614 749.64 391.07 0.8260 17.84

Table 4

Effect of TL1TH1

UHAH ¼ 1 kW/K _mHCPH ¼ 0:2 kW/K TH1 ¼ 523 K

TH1 (K) TL1 (K) TL1TH1

Tw (K) Tc (K) Pmax (kW) gopt (%)

358 298 0.832 342.27 312.33 0.1372 8.78

400 303 0.757 374.00 325.60 0.335 13.00

523 308 0.589 462.20 354.65 1.4036 23.25

643 318 0.494 547.46 385.15 2.816 29.70

663 323 0.487 562.83 392.87 3.00 30.20

740 340 0.459 620.67 420.75 3.809 32.20

783 353 0.450 654.12 439.48 4.207 33.00

Table 5

Effect of _mHCPH

TH1 ¼ 523 K TL1 ¼ 300 K UH1AH ¼ 1 kW/K

_mHCPH

(kW/K)

Tw(K)

Tc(K)

TH2

(K)

TL2(K)

QH

(kW)

QL

(kW)

Pmax

(kW)

gopt(%)

0.2 459.5 348.05 459.92 347.72 12.616 9.54 1.528 24.26

0.4 459.5 348.05 464.712 344.105 23.315 17.64 2.825 24.26

0.6 459.5 348.05 471.49 338.97 30.906 23.38 3.745 24.26

0.8 459.5 348.05 471.49 338.97 30.906 23.38 3.745 24.26

1.0 459.5 348.05 477.069 334.28 36.248 27.42 4.392 24.26

1.2 459.5 348.05 487.09 327.167 43.092 32.60 5.221 24.26

1.4 459.5 348.05 490.58 324.527 45.388 34.33 5.500 24.26

Table 6

Effect of UHAH

TH1 ¼ 523 K TL1 ¼ 300 K UHAH ¼ 1 kW/K

UHAH

(kW/K)

Tw(K)

Tc(K)

TH2

(K)

TL2(K)

QH

(kW)

QL

(kW)

Pmax

(kW)

gopt(%)

1 459.5 348.05 459.5 347.72 12.61 9.544 1.5288 24.26

2 459.5 348.05 459.5 348.04 12.69 9.608 1.539 24.26

3 459.5 348.05 459.5 348.049 12.70 9.609 1.5392 24.26

4 459.5 348.05 459.5 348.049 12.70 9.609 1.5392 24.26

5 459.5 348.05 459.5 348.05 12.70 9.61 1.5392 24.26

6 459.5 348.05 459.5 348.05 12.70 9.61 1.5392 24.26

7 459.5 348.05 459.5 348.05 12.70 9.61 1.5392 24.26


are extended to higher temperature-ranges. This efficiency affords a much more

accurate estimation of real heat-engines than does the classical Carnot efficiency.

Therefore, the formulae derived in this paper serve as a better guide to observable

performance than the Carnot relation.

60

50

40

30

20

10

observed

opt

C

3 4 5 621Temperature Ratio T

H1 / T

L1

Eff

icie

ncy

(%)

Fig. 3. Optimal, observed and Carnot efficiency data vs. TH1=TL1.

30

25

20

15

10

5

60 80 100 120

Opt

imal

eff

icie

ncy

(%)

140 160 180 200

Temperature T (˚C)H1

Fig. 4. Variation of optimal efficiency with initial temperature of the heat source.


4. Results and discussion

For the numerical appreciation of the results, the power output and thermal

efficiency have been computed for a typical set of operating conditions.

TH1 ¼ 358–783 K; UHAH ¼ 1–7 kW=K;

TL1 ¼ 298–333 K; _mHCPH ¼ 0:2–1:0 kW=K:

We used the opportunity offered by the present methodology to survey some of

the efficiency information available on power plants built during the last few decades


(Table 1). It was seen that the observed efficiency data on well-designed real power

plants come closer to the efficiency obtained at maximum power in Eq. (32) and is

shown in Fig. 3 .

The effect of the variation of each of these paramenters on the power output and

thermal efficiency is studied while keeping the other parameters constant. The results

obtained are shown in Tables 2–6. It can be seen from these results, that the poweroutput and thermal efficiency both increase as the inlet source temperature TH1 in-

creases, as shown in Table 2. But on the other hand, if we increase TL1, the inlet sinktemperature, then both the power output and thermal efficiency decrease as shown in

Table 3 . We have also studied the effect of temperature ratio TL1=TH1 and it is seen

that, as this ratio decreases, both the power output and thermal efficiency increase as

shown in Table 4.

It was also found that the cycle is sensitive to the heat capacity rates of the ex-

ternal heating and cooling fluids. As the thermal capacitance ( _mHCPH) of the heatsource increases, the power output increases significantly but thermal efficiency re-

mains constant as shown in Table 5. It was further observed that the effect of thermal

conductance of the heating fluid (UHAH) on both power output and thermal effi-

ciency is insignificant, as shown in Table 6. Optimal values of the operating fluid-

temperatures at which the cycle should operate for maximum power output are also

shown in the results and it was observed that the optimal temperature (Tw) is lowerthan TH1 and the optimal temperature Tc is higher than TL1.

5. Conclusion

The degree of thermodynamic imperfection of a heat engine can be assessed based

on a very simple model that considers external irreversibilities due to finite-time heat-

transfer between the hot and cold end heat-exchangers and the source/sink reser-

voirs. If the designer maximizes the instantaneous power-output, then there are two

important trade-offs to keep in mind: the optimal temperature of heating and coolingfluids, and the optimal balance between the heat conductance and heat capacitance

of the hot and cold heat exchangers. Using the finite-time optimization method, the

power output has been maximized with the above two trade-offs, and the expression

for efficiency has been derived at maximum power. It was found that the efficiency

evaluated from the formula developed here is much closer to that actually observed

in well-designed real heat-engines. The formula developed gives an accurate upper

bound for the performances of real heat-engines.

Overall, the study draws attention to the opportunity for using the most basicthermodynamic and heat-transfer modeling and analysis in order to describe the

irreversible operation of admittedly complicated engineering-systems.

References

[1] Novikov II. The efficiency of atomic power stations. J Nucl Energy 1958;11(5):125–8.


[2] Chambadal P. Les Centrales nucleaires. Paris: Arman colin; 1957. p. 41–58.

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