PPLIED

A

Applied Energy 82 (2005) 181–195

www.elsevier.com/locate/apenergy

ENERGY

Power optimization of an endoreversible closedintercooled regenerated Brayton-cycle coupled to

variable-temperature heat-reservoirs

Wenhua Wang a, Lingen Chen a,*, Fengrui Sun a, Chih Wu b

a Postgraduate School, Naval University of Engineering, Wuhan 430033, PR Chinab Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA

Accepted 20 August 2004

Available online 21 December 2004

Abstract

In this paper, in the viewpoint of finite-time thermodynamics and entropy-generation min-

imization are employed. The analytical formulae relating the power and pressure-ratio are

derived assuming heat-resistance losses in the four heat-exchangers (hot- and cold-side heat

exchangers, the intercooler and the regenerator), and the effect of the finite thermal-capacity

rate of the heat reservoirs. The power optimization is performed by searching the optimum

heat-conductance distributions among the four heat-exchangers for a fixed total heat-exchan-

ger inventory, and by searching for the optimum intercooling pressure-ratio. When the optimi-

zation is performed with respect to the total pressure-ratio of the cycle, the maximum power is

maximized twice and a �double-maximum� power is obtained. When the optimization is per-

formed with respect to the thermal capacitance rate ratio between the working fluid and the

heat reservoir, the double-maximum power is maximized again and a thrice-maximum power

is obtained. The effects of the heat reservoir�s inlet-temperature ratio and the total heat-exchan-

ger inventory on the optimal performance of the cycle are analyzed by numerical examples.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Finite-time thermodynamics; Brayton cycle; Intercooled; Regenerated; Power optimization

0306-2619/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apenergy.2004.08.007

* Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709.

E-mail addresses: lgchenna@yahoo.com, lingenchen@hotmail.com (L. Chen).

Nomenclature

C thermal-capacity rate (kW/K)

E effectivenesses of heat exchangers

k principal specific-heats ratio

N number of heat-transfer units

P power (kW)

p pressure (MPa)Q the rate at which heat is transferred (kJ)

T temperature (K)

U heat conductance (kW/K)

u heat-conductance distributiion

g efficiency

p total pressure-ratio

p1 intercooling pressure-ratio

s1 cycle heat-reservoir temperature-ratios2 cooling fluid in the intercooler and the cold-side heat-reservoir of tem-

peratures ratio

Subscripts

H, h hot-side heat-exchanger

I, i intercooler

in inlet

L, I cold-side heat-exchangermax maximum

max, 2 double maximum

opt optimum

out outlet�Pmax maximum dimensionless power of the cycle�Pmax;2 double maximum dimensionless power of the cycle

R, r regenerator

T totalwf working fluid

l, 2, 3, 4, 5, 6, 7, 8 state points

182 W. Wang et al. / Applied Energy 82 (2005) 181–195

1. Introduction

Since the finite-time thermodynamics (FTT, or endoreversible thermodynamics,

or entropy-generation minimization (EGM), or thermodynamic optimization) was

introduced [1–3], much work has been undertaken on the performance analysis

and optimization of finite-time processes and finite-size devices [4–11]. The FTT per-

formance of closed [12–25,2,27–29] or open [30,31], simple [12–15,22–25,27,28,30],

W. Wang et al. / Applied Energy 82 (2005) 181–195 183

regenerated [16–21,29,31], and intercooled [26], endoreversible [12,13,24,26,27] or

irreversible [14–23,25,28–31] Brayton-cycles has been analyzed and optimized for

the power, specific power, power density, efficiency, and ecological optimization

objectives with heat-transfer irreversibility and/or internal irreversibilities [12–31].

On the basis of that research Chen et al. [32,33] and Wang et al. [34] analyzed theperformance of endoreversible [32] and irreversible [33,34] intercooled and regener-

ated Brayton-cycles coupled constant-temperature [33] and variable-temperature

[32,34] heat-reservoirs, and derived the power and efficiency expressions for the cy-

cles. The further step of this paper beyond that in [32–34] is to optimize the power of

an endoreversible closed intercooled regenerated Brayton-cycle coupled to variable-

temperature heat-reservoirs using FTT by searching the optimum heat-conductance

distributions among the four heat-exchangers (the hot- and cold-side heat-exchang-

ers, the intercooler and the regenerator) for a fixed total heat-exchanger inventory,searching for the optimal intercooling pressure-ratio and the optimal total pres-

sure-ratio, and searching for the optimal matching of the thermal capacitance rate

ratio between the working, fluid and the heat reservoir. The effects of the heat-reser-

voir inlet-temperature ratio and the total heat-exchanger inventory on the optimal

performance of the cycle are analyzed by detailed numerical examples.

2. Theoretical model

An endoreversible closed intercooled regenerated Brayton-cycle 1–2–3–4–5–6–1

coupled to variable-temperature heat reservoirs is shown in Fig. 1. The high-temper-

ature (hot-side) heat-reservoir is considered to have a thermal capacity rate CH and

the inlet and the outlet temperatures of the heating fluid are THin and THout, respec-

tively. The low-temperature (cold-side) heat-reservoir is considered to have a thermal

capacity rate CL and the inlet and the outlet temperatures of the cooling fluid are

TLin and TLout, respectively. The cooling fluid in the intercooler is considered to havea thermal-capacity rate CI and the inlet and the outlet temperatures are TLin and

TIout, respectively. Processes 1 ! 2 and 3 ! 4 are isentropic adiabatic compression

in the low- and high-pressure compressors, while the process 5! 6 is isentropic

expansion in the turbine. Process 2 ! 3 is an isobaric intercooling in the intercooler.

Process 4 ! 7 is an isobaric absorbed-heat process and process 6 ! 8 an isobaric

evolved heat-process in the regenerator. Process 7 ! 5 is an isobaric absorbed-heat

process in the hot-side heat exchanger and process 8 ! 1 an isobaric evolved heat-

process in the cold-side heat-exchanger.Assuming that the working fluid used in the cycle is an ideal gas with a constant

thermal-capacity rate (mass-flow rate and specific heat product) Cwf, the heat

exchangers between the working fluid and the heat reservoirs, the regenerator and

the intercooler are counter-flow and the heat conductances (heat-transfer surface

area and heat-transfer coefficient product) of the hot- and cold-side heat exchangers,

the regenerator and the intercooler are UH, UL, UR and UI respectively. According to

the properties of the working fluid, the heat-transfer processes, the properties of the

heat reservoirs and the theory of heat-exchangers, the rate (QH) at which heat is

Fig. 1. T–S diagram of an endoreversible intercooled regenerated Brayton-cycle coupled to variable-

temperature reservoirs.

184 W. Wang et al. / Applied Energy 82 (2005) 181–195

transferred from the heat source to the working fluid, the rate (QL) at which heat is

rejected from the working fluid to the heat sink, the rate (QR) of heat regenerated inthe regenerator, and the rate (QI) of heat exchanged in the intercooler are, respec-

tively, given by

QH ¼ CwfðT 5 � T 7Þ ¼ UH

ðTHin � T 5Þ � ðTHout � T 7Þln½ðTHin � T 5Þ=ðTHout � T 7Þ�

¼ CHðTHin � THoutÞ ¼ CHminEHðTHin � T 7Þ; ð1Þ

QL ¼ CwfðT 8 � T 1Þ ¼ UL

ðT 8 � T LoutÞ � ðT 1 � T LinÞln½ðT 8 � T LoutÞ=ðT 1 � T LinÞ�

¼ CLðT Lout � T LinÞ

¼ CLminELðT 8 � T LinÞ; ð2Þ

QR ¼ CwfðT 7 � T 4Þ ¼ CwfðT 6 � T 8Þ ¼ CwfERðT 6 � T 4Þ ð3Þand

QI ¼ CwfðT 2 � T 3Þ ¼ U I

ðT 2 � T IoutÞ � ðT 3 � T IinÞln½ðT 2 � T IoutÞ=ðT 3 � T IinÞ�

¼ CIðT Iout � T IinÞ

¼ CIminEIðT 2 � T IinÞ; ð4Þ

where EH, EL, ER and EI are the component (the hot-and cold-side heat-exchangers,

the regenerator and the intercooler) effectivenesses, and are defined as

EH ¼ 1� exp½�NHð1� CHmin=CHmaxÞ�1� ðCHmin=CHmaxÞ exp½�NHð1� CHmin=CHmaxÞ�

; ð5Þ

W. Wang et al. / Applied Energy 82 (2005) 181–195 185

EL ¼ 1� exp½�NLð1� CLmin=CLmaxÞ�1� ðCLmin=CLmaxÞ exp½�NLð1� CLmin=CLmaxÞ�

; ð6Þ

ER ¼ NR=ðNR þ 1Þ ð7Þ

and

EI ¼1� exp½�N Ið1� CImin=CImaxÞ�

1� ðCImin=CImaxÞ exp½�N Ið1� CImin=CImaxÞ�; ð8Þ

where CHmin and CHmax are the smaller and the larger of the two capacitance rates

CH and Cw/f, respectively; CLmin and CLmax are the smaller and the larger of the twocapacitance rates CL and Cwf, respectively; CImin and CImax are the smaller and the

larger of the two capacitance rates CI, and Cwf, respectively. The number of heat

transfer units NH, NL, NR and NI of the hot- and cold-side heat-exchangers, the

regenarator and the intercooler, respectively, are

NH ¼ UH=CHmin; NL ¼ UL=CLmin; NR ¼ UR=Cwf ; N I ¼ U I=CImin; ð9Þwhere

CHmin ¼ minfCH;Cwfg; CHmax ¼ maxfCH;Cwfg; ð10Þ

CLmin ¼ minfCL;Cwfg; CLmax ¼ maxfCL;Cwfg; ð11Þand

CImin ¼ minfCI;Cwfg; CImax ¼ maxfCI;Cwfg: ð12ÞThe cycle�s power-output and the cycle efficiency are defined as

P ¼ QH � QL � QI; ð13Þ

g ¼ 1� ðQL þ QIÞ=QH: ð14Þ

Defining the working-fluid�s isentropic temperature-ratios x and y for the low-

pressure compressor and total compression-process gives

x ¼ T 2=T 1 ¼ ðP 2=P 1Þm ¼ pm1 ; ð15Þ

y ¼ T 5=T 6 ¼ ðP 5=P 6Þm ¼ pm; ð16Þ

where m = (k � 1)/k (k is the ratio of principal specific heats), p1 is the intercooling

pressure ratio and p is the total pressure-ratio. The temperature ratio in the high-

pressure compressor can be written as

y=x ¼ P 4=P 3 ¼ T 4=T 3: ð17Þ

Combining Eqs. (1)–(17), one can find the power output and the efficiency of the

endoreversible intercooled regenerated Brayton-cycle coupled to the variable-

temperature heat reservoirs. The dimensionless power �P ð�P ¼ P=ðCLT LinÞÞ and the

efficiency g are

186 W. Wang et al. / Applied Energy 82 (2005) 181–195

ð18Þ

ð19Þ

Fig. 2. Optimal dimensionless power versus total heat-exchanger inventory and intercooling pressure-

ratio.

Fig. 3. Maximum dimensionless power versus heat-reservoir inlet temperature-ratio and total pressure-

ratio.

W. Wang et al. / Applied Energy 82 (2005) 181–195 187

where s = THin/TLin is the cycle�s heat-reservoir inlet temperature ratio, s2 = TLin/

TLin is the inlet temperature ratio of the cooling fluid in the intercooler and the

cold-side heat-reservoir, a1 = CHmin/Cwf, a2 = CLmin/Cwf and a3 = CImin/Cwf.

3. Power optimization

Eq. (18) indicates that the cycle�s dimensionless-power �P is a function of the total

pressure-ratio p, intercooling pressure-ratio p1, heat conductance UH of the hot-side

heat-exchanger, heat conductance UL of the cold-side heat-exchanger, heat conduc-

tance UR of the regenerator, heat conductance UI of the intercooler, and the thermal

capacitance rate Cwf of the working fluid for the fixed boundary conditions of the

Fig. 4. Efficiency at maximum dimensionless power versus heat-reservoir inlet-temperature ratio and total

pressure-ratio.

Fig. 5. Maximum dimensionless power versus corresponding efficiency and heat reservoir inlet-

temperature ratio.

188 W. Wang et al. / Applied Energy 82 (2005) 181–195

cycle. For the fixed p, p1 and Cwf,there exist a group of optimal distributions among

UH, UL, UR and UI, which lead to the optimal dimensionless-power �P opt, as the total

heat-exchanger inventory (UT = UH + UL + UR + Ul) is fixed. For the fixed p and

Cwf, there exist an optimal intercooling pressure-ratio and a group of optimal distri-

butions among UH, UL, UR and UI, as UT is fixed, that lead to the maximum dimen-sionless-power Pmax. Optimizing the maximum power with respect to the total

pressure-ratio p for the fixed Cwf, one can obtain the �double-maximum� power�Pmax;2. When the optimization is performed with respect to the thermal-capacitance

rate ratio between the working fluid and the heat reservoir, the �double-maximum�power is maximized again and a �thrice-maximum� power is obtained.

For the fixed total heat-exchange inventory (UT = UH + UL + UR + UI), defining

uh ¼ UH=UT; ul ¼ UL=UT; ui ¼ U I=UT and ur ¼ 1� uh � u1 � ui;

ð20Þ

Fig. 6. Optimum intercooling pressure-ratio at maximum dimensionless power versus total pressure-ratio

and heat-reservoir inlet-temperature ratio.

Fig. 7. Heat-conductance distribution of hot-side exchanger at maximum power versus total pressure-

ratio and heat-reservoir inlet-temperature ratio.

W. Wang et al. / Applied Energy 82 (2005) 181–195 189

leads to

UH ¼ uhUT; UL ¼ u1UT; U I ¼ uiUT and UR ¼ ð1� uh � ul � uiÞUT:

ð21ÞAdditionally, one has the constraints

0 < uh þ ul < 1; 0 < uh þ ui < 1; 0 < ul þ ui < 1 and

0 < uh þ ul þ ui < 1:ð22Þ

The power optimization is performed by numerical calculations, and the compu-tational program is integrated with the optimization toolbox of MATLAB. In the

calculations, k = 1.4, TLin = 300 K, s2 = 1, and CH = CL = CI = 1.2 kW/K are set.

Fig. 8. Heat conductance distribution of cold-side exchanger at maximum power versus total pressure-

ratio and heat-reservoir inlet-temperature ratio.

Fig. 9. Heat-conductance distribution of intercooler at maximum power versus total pressure-ratio and

heat-reservoir inlet-temperature ratio.

190 W. Wang et al. / Applied Energy 82 (2005) 181–195

Fig. 2 shows the characteristic of the optimal dimensionless-power �P opt versus to-

tal heat-exchanger inventory UT and intercooling pressure-ratio p1. The optimal

dimensionless-power �P opt has a maximum value, i.e. the maximum dimensionless-

power �Pmax, with respect to p1. For a fixed p1, one can find that the increment of�P opt is less as the total heat-conductance UT increases.

Figs. 3–9 show the characteristics of the maximum dimensionless power �Pmax and

its corresponding parameters, i.e., efficiency g�Pmax, optimal intercooling pressure-ratio

ðp1Þ�Pmax, optimal heat-conductance distribution ðuhÞ�Pmax

of the hot-side heat-exchan-

ger, optimal heat-conductance distribution ðulÞ�Pmaxof the cold-side heat-exchanger

and optimal heat-conductance distribution ðuiÞ�Pmaxof the intercooler, versus the total

pressure-ratio p and heat reservoir-inlet temperature-ratio s1.

Fig. 10. �Double-maximum� dimensionless power and its corresponding efficiency versus heat-reservoir

inlet-temperature ratio.

Fig. 11. Total pressure-ratio at �double-maximum� dimensionless power versus heat-reservoir inlet-

temperature ratio.

W. Wang et al. / Applied Energy 82 (2005) 181–195 191

Fig. 2 shows that the characteristic curve of �Pmax versus p is parabolic like. As pincreases, �Pmax will increase to its peak value, i.e. a �double-maximum� dimensionless

power �Pmax;2, and then decrease smoothly: �Pmax increases as the heat reservoir inlet

temperature ratio s1 increases. Fig. 5 shows that the characteristic curve of �Pmax ver-

sus its corresponding efficiency g�Pmaxis part loop-shaped that is similar to that of a

generalized irreversible Carnot heat-engine [35], and it indicates the optimal work-

ing-range of the cycle. Fig. 6 shows that ðp1Þ�Pmaxis an increasing function of p, while

a decreasing function of s1. Fig. 7 shows that the characteristic curve of ðuhÞ�Pmaxver-

sus p is parabolic like. Fig. 8 shows that when s1 is larger than a specified value (e.g.,

s1 P 5), ðulÞ�Pmaxwill increase first, and then decrease as p increases; if s1, is smaller

than some specified value (e.g., s1 6 3), ðulÞ�Pmaxwill decrease monotonically;

Fig. 12. �Double-maximum� dimensionless power and its corresponding efficiency versus total heat-

exchanges inventory.

Fig. 13. Total pressure ratio at �double-maximum� dimensionless power versus total heat-exchanger

inventory.

192 W. Wang et al. / Applied Energy 82 (2005) 181–195

ðulÞ�Pmaxincreases with increases in s1. Fig. 9 shows that ðulÞ�Pmax

is an increasing func-

tion of p, but a decreasing function of s1.Figs. 10 and 11 illustrate the characteristics of the �double-maximum� dimension-

less power �Pmax;2 and its corresponding efficiency g�Pmax;2and optimal total pressure-

ratio p�Pmax;2versus the heat reservoir inlet-temperature ratio s1. Figs. 12 and 13

illustrate the characteristics of �Pmax;2; g�Pmax;2and p�Pmax;2

versus the total heat-

conductance UT. It can be seen that �Pmax;2; g�Pmax;2and p�Pmax;2

are increasing functions

of s1. As UT increases, �Pmax;2 and p�Pmax;2increase, but the increment will be less and

less. The characteristic curve of g�Pmax;2versus UT is parabolic like and it exhibits an

optimal working range of the cycle, the power output is larger and the corresponding

efficiency is higher.

Fig. 14. �Double-maximum� dimensionless-power and its corresponding efficiency versus thermal-

capacitance ratio between working fluid and heat reservoir.

Fig. 15. Optimum total pressure-ratio at �double-maximum� dimensionless power versus thermal-

capacitance rate ratio between working fluid and heat reservoir.

W. Wang et al. / Applied Energy 82 (2005) 181–195 193

Figs. 14 and 15 illustrate the characteristics of the �double-maximum� dimension-

less-power �Pmax;2 and its corresponding efficiency g�Pmax;2and optimal total pressure-

ratio p�Pmax;2versus the thermal-capacitance rate ratio Cwf/CL between the working

fluid and cold-side heat-reservoir. It can be seen that �Pmax;2 has an additional max-

imum with respect to Cwf/CL. For the conditions given above, the optimal ther-mal-capacitance rate ratio between the working fluid and cold-side heat-reservoir

is about 0.8 for the �thrice-maximum� power. Similar characteristic can be found be-

tween g�Pmax;2and Cwf/CL. As Cwf/CL increases, p�Pmax;2

decreases first, and then in-

creases sharply.

4. Conclusion

Finite-time thermodynamics has been applied to optimize the power of an endo-

reversible closed intercooled regenerated Brayton-cycle coupled to variable-temper-

ature heat-reservoirs. The analysis indicates that there exists a group of optimal

values for p1, UH, UL, UR and UI which lead to the maximum dimensionless-power.

Furthermore, there exists an optimal total pressure-ratio that leads to a �double-max-

imum� power-output. The power ð�P opt; �Pmax or �Pmax;2Þ increases with the increases in

the cycle�s heat-reservoir temperature-ratio and the total heat-exchanger inventory.

When UT is larger than a particular value, the increment of the cycle power becomesless and less. The �double maximum� power has an additional maximum with respect

to the thermal-capacitance rate ratio between the working fluid and heat reservoir.

An optimal cycle working-range, i.e. high power output and high efficiency, can

be obtained according to the characteristics of the maximum power and/or the dou-

ble maximum power and its corresponding efficiency. The analysis and optimization

presented herein may provide some guidelines for power-optimization design for the

closed gas-turbine power-plants.

Acknowledgements

This paper is supported by the Foundation for the Authors of Nationally Excel-

lent Doctoral Dissertations of PR China (Project No. 200136) and the Natural Sci-

ence Foundation of the Naval University of Engineering of PR China (Project No.

HGDJJ03016).

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