performance analysis for an irreversible variable temperature heat reservoir closed intercooled...

Energy Conversion and Management 44 (2003) 2713–2732www.elsevier.com/locate/enconman

Performance analysis for an irreversible variabletemperature heat reservoir closed intercooled

regenerated Brayton cycle

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

a Faculty 306, Naval University of Engineering, Wuhan 430033, PR Chinab Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA

Received 26 September 2002; accepted 28 January 2003

Abstract

In this paper, the theory of finite time thermodynamics is used in the performance analysis of an irre-

versible closed intercooled regenerated Brayton cycle coupled to variable temperature heat reservoirs. The

analytical formulae for dimensionless power and efficiency, as functions of the total pressure ratio, the

intercooling pressure ratio, the component (regenerator, intercooler, hot and cold side heat exchangers)

effectivenesses, the compressor and turbine efficiencies and the thermal capacity rates of the working fluid

and the heat reservoirs, the pressure recovery coefficients, the heat reservoir inlet temperature ratio, and thecooling fluid in the intercooler and the cold side heat reservoir inlet temperature ratio, are derived. The

intercooling pressure ratio is optimized for optimal power and optimal efficiency, respectively. The effects of

component (regenerator, intercooler and hot and cold side heat exchangers) effectivenesses, the compressor

and turbine efficiencies, the pressure recovery coefficients, the heat reservoir inlet temperature ratio and the

cooling fluid in the intercooler and the cold side heat reservoir inlet temperature ratio on optimal power and

its corresponding intercooling pressure ratio, as well as optimal efficiency and its corresponding inter-

cooling pressure ratio are analyzed by detailed numerical examples. When the heat transfers between the

working fluid and the heat reservoirs are executed ideally, the pressure drop losses are small enough to beneglected and the thermal capacity rates of the heat reservoirs are infinite, the results of this paper replicate

those obtained in recent literature.

� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Finite time thermodynamics; Brayton cycle; Intercooled; Regenerated; Irreversible

* Corresponding author.

E-mail address: [email protected] (L. Chen).

0196-8904/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0196-8904(03)00046-3

mail to: [email protected]

Nomenclature

CH, CL thermal capacity rates of high and low temperature heat reservoirsCI thermal capacity rate of cooling fluid in intercoolerCwf thermal capacity rate of working fluid (mass flow rate and specific heat product)D1, D2 pressure recovery coefficientsEH1, EL1 effectivenesses of hot and cold side heat exchangersER effectiveness of regeneratorEI1 effectiveness of intercoolerNH1, NL1 number of heat transfer units of hot and cold side heat exchangersNR number of heat transfer units of regeneratorNI1 number of heat transfer units of intercoolerk ratio of specific heatsp1, p2, p3, p4, p5, p6 pressures at working states of 1, 2, 3, 4, 5, 6P opt optimal dimensionless powerPmax maximum dimensionless powerQH rate at which heat is transferred from heat source to working fluidQL rate at which heat is transferred from working fluid to heat sinkQR rate of heat regenerated in the regeneratorQI rate of heat rejected from working fluid to cooling fluid in intercoolerT1, T2, T2s, T3, T4, T4s, T5, T6, T6s, T7, T8 temperatures at states of 1, 2, 2s, 3, 4, 4s, 5, 6, 6s, 7, 8TH in, THout inlet and outlet temperatures of heating fluidTL in, TL out inlet and outlet temperatures of cooling fluidTI in, TI out inlet and outlet temperatures of cooling fluid in intercoolerUH, UL conductances of hot and cold side heat exchangers (heat transfer surface area and

heat transfer coefficient product)UR conductance of regeneratorUI conductance of intercoolerx working fluid isentropic temperature in low pressure compressory working fluid isentropic temperature for whole compression process1, 2, 2s, 3, 4, 4s, 5, 6, 6s, 7, 8 working states

Greeks

gc, gt compressor and turbine efficienciesgopt optimal efficiencygmax maximum efficiencyp total pressure ratiop1 intercooling pressure ratioðp1ÞPopt

intercooling pressure ratio corresponding to optimal dimensionless powerðp1Þgopt intercooling pressure ratio corresponding to optimal efficiencys1 cycle heat reservoir inlet temperature ratios2 cooling fluid in intercooler and cold side heat reservoir inlet temperature ratio

2714 W. Wang et al. / Energy Conversion and Management 44 (2003) 2713–2732

W. Wang et al. / Energy Conversion and Management 44 (2003) 2713–2732 2715

1. Introduction

Since the theory of finite time thermodynamics (FTT) or entropy generation minimization(EGM) was advanced [1–3], much work has been performed for the performance analysis andoptimization of finite time processes and finite size devices [4–10]. Many achievements have beenacquired on the performance analysis of Brayton cycles using FTT [11]. Leff [12] found the simplereversible Brayton cycle�s efficiency at maximum work output was g ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiTL=TH

p. Bejan [13]

took the entropy generation rate as the optimization objective and analyzed the performance ofan endoreversible simple Brayton cycle and proved the power output reaches its maximum whenthe entropy generation rate of the cycle is minimum. Chen and co-workers [14–22] analyzed andoptimized the power, the power density and the efficiency for simple and regenerated, endore-versible and irreversible Brayton cycles coupled to constant and variable temperature heat reser-voirs, deduced the general analytical formulae of the cycle power, the power density and theefficiency and analyzed the effects of the internal and the external irreversibilities on the cyclecharacteristics. Ibrahim et al. [23] examined the optimal power output of an endoreversible simpleBrayton cycle coupled to constant and variable temperature heat reservoirs. Roco et al. [24]analyzed the maximum power and its corresponding efficiency as well as the maximum efficiencyand its corresponding power of a regenerated Brayton cycle based on Ref. [16]. Feidt [25] analyzedthe performance of an irreversible closed regenerated Brayton cycle by taking account of heatresistance and heat leak. Cheng and Chen [26] built a model for an endoreversible intercooledBrayton cycle, evaluated the maximum power and its corresponding efficiency and investigatedthe effects of the intercooled process on the power output and efficiency of the cycle.

A further step of this paper is to study the performance of an irreversible closed intercooled andregenerated Brayton cycle coupled to variable temperature heat reservoirs using FTT. Analyticalformulae about dimensionless power and efficiency are derived. The effects of component (re-generator, intercooler and hot and cold side heat exchangers) effectivenesses, compressor andturbine efficiencies, pressure recovery coefficients, heat reservoir inlet temperature ratio andcooling fluid in the intercooler and cold side heat reservoir inlet temperature ratio on optimalpower and its corresponding intercooling pressure ratio, as well as optimal efficiency and itscorresponding intercooling pressure ratio are analyzed by detailed numerical examples. Espe-cially, the intercooling pressure ratio is optimized for optimal power and optimal efficiency,respectively.

2. Cycle analysis

An irreversible closed intercooled regenerated Brayton cycle 1–2–3–4–5–6–1 coupled to variabletemperature heat reservoirs is shown in Fig. 1. Processes 1–2 and 3–4 are non-isentropic adiabaticcompression processes in the low and high pressure compressors, while process 5–6 is the non-isentropic expansion process in the turbine. Process 2–3 is an isobaric intercooling process in theintercooler. Process 4–7 is an isobaric absorbed heat process, and process 6–8 is an isobar evolvedheat process in the regenerator. Process 7–5 is an isobaric absorbed heat process in the hot side heatexchanger, and process 8–1 is an isobar evolved heat process in the cold side heat exchanger.Processes 1–2s, 3–4s and 5–6s are isentropic adiabatic processes representing the processes in the

Fig. 1. The real intercooled regenerated Brayton cycle.


ideal low and high pressure compressors and ideal turbine, respectively. The pressure drop Dp inthe piping is reflected by using pressure recovery coefficients, that is,

D1 ¼ p5=p4; D2 ¼ p1=p6 ð1Þ
Assuming the low and high pressure compressors have the same efficiency gc and the turbine
efficiency is gt, and gc and gt reflect the irreversibility in the non-isentropic compression andexpansion processes, respectively, leads to

gc ¼ ðT2s � T1Þ=ðT2 � T1Þ ¼ ðT4s � T3Þ=ðT4 � T3Þ; gt ¼ ðT5 � T6Þ=ðT5 � T6sÞ ð2Þ
Assuming that the working fluid used in the cycle is an ideal gas, the heat exchangers between
the working fluid and the heat reservoirs, the regenerator and the intercooler are counter flow,according to the properties of the heat transfer processes, heat reservoirs, working fluid and heatexchangers, one can obtain

QH ¼ UH

ðTH in � T5Þ � ðTHout � T7Þln½ðTH in � T5Þ=ðTHout � T7Þ�

¼ CHðTH in � THoutÞ ¼ CwfðT5 � T7Þ

¼ CHminEH1ðTH in � T7Þ ð3Þ

QL ¼ UL

ðT8 � TLoutÞ � ðT1 � TL inÞln½ðT8 � TLoutÞ=ðT1 � TL inÞ�

¼ CLðTLout � TL inÞ ¼ CwfðT8 � T1Þ

¼ CLminEL1ðT8 � TL inÞ ð4Þ

QR ¼ CwfðT7 � T4Þ ¼ CwfðT6 � T8Þ ¼ CwfERðT6 � T4Þ ð5Þ

QI ¼ UI

ðT2 � TI outÞ � ðT3 � TI inÞln½ðT2 � TI outÞ=ðT3 � TI inÞ�

¼ CIðTI out � TI inÞ ¼ CwfðT2 � T3Þ

¼ CIminEI1ðT2 � TI inÞ ð6Þ


where EH1, EL1, ER and EI1 are defined as

EH1 ¼ f1� exp½�NH1ð1�CHmin=CHmaxÞ�g=f1� ðCHmin=CHmaxÞ exp½�NH1ð1�CHmin=CHmaxÞ�gð7Þ

EL1 ¼ f1� exp½�NL1ð1� CLmin=CLmaxÞ�g=f1� ðCLmin=CLmaxÞ exp½�NL1ð1� CLmin=CLmaxÞ�gð8Þ

ER ¼ NR=ðNR þ 1Þ ð9Þ

EI1 ¼ f1� exp½�NI1ð1� CImin=CImaxÞ�g=f1� ðCImin=CImaxÞ exp½�NI1ð1� CImin=CImaxÞ�gð10Þ

where CHmin and CHmax are the smaller and the larger of the two capacitance rates CH and Cwf ,respectively;CLmin and CLmax are the smaller and the larger of the two capacitance rates CL andCwf , respectively; and CImin and CImax are the smaller and the larger of the two capacitance rates CI

and Cwf , respectively; and NH1, NL1, NR and NI1 are defined as

NH1 ¼ UH=CHmin; NL1 ¼ UL=CLmin; NR ¼ UR=Cwf ; NI1 ¼ UI=CImin ð11Þ

CHmin ¼ minfCH;Cwfg; CHmax ¼ maxfCH;Cwfg ð12Þ

CLmin ¼ minfCL;Cwfg; CLmax ¼ maxfCL;Cwfg ð13Þ

CImin ¼ minfCI;Cwfg; CImax ¼ maxfCI;Cwfg ð14Þ

Then, the cycle power output and the cycle efficiency can be written as

P ¼ QH � QL � QI ¼ CwfðT1 � T2 þ T3 þ T5 � T7 � T8Þ ð15Þ

g ¼ P=QH ¼ 1� ðT8 � T1 þ T2 � T3Þ=ðT5 � T7Þ ð16Þ

According to the knowledge of thermodynamics, one can obtain

x ¼ T2s=T1 ¼ ðp2=p1Þm ¼ pm1 ð17Þ

T5=T6s ¼ ðp5=p6Þm ¼ yD�1; D ¼ ðD1D2Þ�m ð18Þ

where m ¼ ðk � 1Þ=k.The second law of thermodynamics requires

T1T3T5 ¼ T2sT4sT6s ð19Þ

Combining Eqs. (1)–(6) and Eqs. (15)–(19) gives the power output and the efficiency of thecycle. The dimensionless power P ðP ¼ P=ðCLTL inÞÞ and the efficiency g, respectively, are as fol-lows:


P ¼ f1�

� b1b2x�1g�2c ð1� a2EL1Þð1� a3EI1Þ½ER þ b3ð1� 2ERÞ� � b3ER

� b3ð1� ERÞ ½a2EL1 þ b2g�1c ð1� a2EL1Þa3EI1�ga1EH1s1

� f½1� b3ð1� a1EH1ÞER�½b2g�1c a3 EI1 � 1� þ b1b2x�1g�2

c ð1� a3EI1Þ� ½ER þ b3ð1� a1EH1Þð1� 2ERÞ þ ð1� ERÞa1 EH1�ga2EL1 þ f1� b3ð1� a1EH1ÞER

� b1x�1g�1c ð1� ERÞa1EH1 � b1x�1g�1

c ½ER þ b3ð1� a1EH1Þð1� 2ERÞ�

� ½a2EL1 þ b2g�1c ð1� a2EL1Þ�ga3EI1s2

�= 1�

� b3ð1� a1EH1ÞER

� b1b2x�1g�2c ð1� a2EL1Þð1� a3EI1Þ½ER þ b3ð1� a1EH1Þð1� 2ERÞ�

�ð20Þ

g ¼ 1� ½a2EL1

�þ b2g�1

c ð1� a2EL1Þa3EI1�b3ð1� ERÞa1EH1s1

þ f½1� b3ð1� a1EH1Þ ER�½b2g�1c a3EI1 � 1� þ ½ER þ b3ð1� a1EH1Þð1� 2ERÞ�b1b2x�1g�2

c

� ð1� a3 EI1Þga2EL1 þ fb1x�1g�1c a2EL1ER þ b1b3x�1g�1

c ð1� a1EH1Þð1� 2ERÞ

� a2EL1 þ b2g�1c ð1� a2EL1Þ� þ b3ð1� a1EH1ÞER � 1ga3EI1s2

��

= ff1�

� b1b2x�1g�2c ð1� a2EL1Þð1� a3EI1Þ½ER þ b3ð1� 2ERÞ� � b3ERgs1

� b1b2 x�1g�2c ð1� a3EI1Þð1� ERÞa2EL1 � b1x�1g�1

c ð1� ERÞa3EI1s2ga1EH1

�ð21Þ

where a1 ¼ CHmin=Cwf , a2 ¼ CLmin=Cwf , a3 ¼ CImin=Cwf , b1 ¼ y þ xgc � x, b2 ¼ xþ gc � 1,b3 ¼ 1� gt þ y�1Dgt, s1 ¼ TH in=TL in, and s2 ¼ TI in=TL in.

Using Eqs. (20) and (21), one can analyze the effects of component (regenerator, intercooler andhot and cold side heat exchangers) effectivenesses, compressor and turbine efficiencies, pressurerecovery coefficients, heat reservoir inlet temperature ratio and cooling fluid in the intercooler andcold side heat reservoir inlet temperature ratio on the dimensionless power (P ) and the efficiency(g) of the irreversible intercooled regenerated Brayton cycle.

Eqs. (20) and (21) also include many results of recent literature.If the thermal capacity rates of the heat reservoirs and the cooling fluid in the intercooler are

infinite (CH ¼ CL ¼ CI ! 1), the low and high pressure ratio compressors and the turbineare ideal (gc ¼ gt ¼ 1) and there is no pressure drop in the cycle (D1 ¼ D2 ¼ 1), Eqs. (20) and (21)are the results of an endoreversible closed intercooled regenerated Brayton cycle coupled toconstant temperature heat reservoirs

P ¼ fy2ð1�

� ELÞð1� EIÞER þ y½ð1� ELÞð1� 2ERÞð1� EIÞ � 1�� ð1� ELÞð1� ERÞð1� xEIÞ þ 1gEHs1 þ fy2½ð1� ERÞEH þ ER�ð1� EIÞþ y½ð1� EHÞð1� 2ERÞð1� EIÞ � ð1� xEIÞ� þ ð1� EHÞð1� xEIÞERgEL

þ fy2½ð1� ELÞER þ x�1ð1� ERÞEH þ x�1ELER�þ y½ð1� EHÞð1� 2ERÞð1þ x�1EL � ELÞ � 1� þ ð1� EHÞERgEIs2

�

= y2ð1�

� ELÞð1� EIÞER þ y½ð1� EHÞð1� ELÞð1� 2ERÞð1� EIÞ � 1� þ ð1� EHÞER

�

ð22Þ


g ¼ 1� ½ð1�

� ELÞð1� xEIÞ � 1�ð1� ERÞEHs1 � fy2ð1� EIÞER þ y½ð1� EHÞð1� 2ERÞð1� EIÞ� ð1� xEIÞ� þ ð1� EHÞð1� xEIÞERgEL � fy2ð1� EL þ x�1ELÞER

þ y½ð1� EHÞð1� 2ERÞð1� EL þ x�1ELÞ � 1� þ ð1� EHÞERgEIs2�

= fy2ð1�

� ELÞð1� EIÞER þ y½ð1� ELÞð1� 2ERÞð1� EIÞ � 1� þ ERgEHs1

þ y2ð1� ERÞð1� EIÞEHEL þ y2x�1ð1� ERÞEHEIs2�

ð23Þ

where s1 ¼ TH=TL, s2 ¼ TI=TL, EH ¼ 1� expð�NHÞ, EL ¼ 1� expð�NLÞ, EI ¼ 1� expð�NIÞ,NH ¼ UH=Cwf , NL ¼ UL=Cwf , and NI ¼ UI=Cwf .

If the thermal capacity rates of the heat reservoirs and the cooling fluid in the intercooler areinfinite (CH ¼ CL ¼ CI ! 1), the low and high pressure ratio compressors and the turbine areideal (gc ¼ gt ¼ 1), there is no pressure drop in the cycle (D1 ¼ D2 ¼ 1) and the effectiveness of theregenerator is zero (ER ¼ 0), Eqs. (20) and (21) become the results of an endoreversible closedintercooled Brayton cycle coupled to constant temperature heat reservoirs [26]

P ¼ f1�

� ð1� ELÞð1� EIÞ � y�1½EL þ xð1� ELÞEI�gEHs1

� fðxEI � 1Þ þ yð1� EIÞ½y�1ð1� EHÞ þ EH�gEL þ f1� yx�1EH � x�1ð1� EHÞ� ½EL:þ xð1� ELÞ�gEIs2

�= 1f � ð1� ELÞð1� EIÞð1� EHÞg ð24Þ

g ¼ 1� ½EL

�þ xð1� ELÞEI�y�1EHs1 þ f½xEI � 1� þ ð1� EHÞð1� EIÞgEL

þ fx�1ð1� EHÞ½EL þ xð1� ELÞ� � 1gEIs2�= f½1�

� ð1� ELÞð1� EIÞ�s1� yð1� EIÞEL � yx�1EIs2gEH

�ð25Þ

If the intercooling process is not involved in the cycle (x ¼ 1 and EI ¼ 0), Eqs. (20) and (21)become [17]

P ¼ f1�

� ðyþ gc � 1Þg�1c ð1� a2EL1Þ½ER þ ð1� gt þ y�1DgtÞð1� 2ERÞ�

� ð1� gt þ y�1DgtÞER � ð1� gt þ y�1DgtÞð1�ERÞa2EL1ga1EH1s1

�f½1� ð1� gt þ y�1DgtÞð1� a1EH1ÞER�ða3EI1 � 1Þ þ ðyþ gc � 1Þg�1c

� ER½ þ ð1� gt þ y�1DgtÞð1� a1EH1Þð1� 2ERÞ þ ð1�ERÞa1EH1�ga2EL1

�

= f1�

� ð1� gt þ y�1DgtÞð1� a1EH1ÞER � ðyþ gc � 1Þg�1c ð1� a2EL1Þ

� ER½ þ ð1� gt þ y�1DgtÞð1� a1EH1Þð1� 2ERÞ�gCL

�ð26Þ

g ¼ 1� a2EL1ð1�

� gt þ y�1DgtÞð1� ERÞa1EH1s1 þ f½1þ ð1� gt þ y�1DgtÞð1� a1EH1ÞER�þ ½ER þ ð1� gt þ y�1DgtÞð1� a1EH1Þð1� 2ERÞ�ðy þ gc � 1Þg�1

c ga2EL1

�

= ff1�

� ðy þ gc � 1Þg�1c ð1� a2EL1Þ½ER þ ð1� 2ERÞð1� gt þ y�1DgtÞ�

� ð1� gt þ y�1DgtÞERgs1 � ðy þ gc � 1Þg�1c ð1� ERÞa2EL1ga1EH1

�ð27Þ

If the intercooling process is not involved in the cycle (x ¼ 1 and EI ¼ 0) and there is nopressure drop in the cycle (D1 ¼ D2 ¼ 1), Eqs. (20) and (21) become [15]


P ¼ f1�

� ðy þ gc � 1Þg�1c ð1� a2EL1Þ½ER þ ð1� gt þ y�1gtÞð1� 2ERÞ� � ð1� gt þ y�1gtÞER

� ð1� gt þ y�1gtÞð1� ERÞa2EL1ga1EH1s1 � f½1� ð1� gt þ y�1gtÞð1� a1EH1ÞER�� ða3EI1 � 1Þ þ ðy þ gc � 1Þg�1

c ½ER þ ð1� gt þ y�1gtÞð1� a1EH1Þð1� 2ERÞþ ð1� ERÞa1EH1�ga2EL1

�= f1�

� ð1� gt þ y�1gtÞð1� a1EH1ÞER � ðy þ gc � 1Þg�1c

� ð1� a2EL1Þ½ER þ ð1� gt þ y�1gtÞð1� a1EH1Þð1� 2ERÞ�gCL

�ð28Þ

g ¼ 1� a2EL1ð1�

� gt þ y�1gtÞð1� ERÞa1EH1s1 þ f½1þ ð1� gt þ y�1gtÞð1� a1EH1ÞER�þ ½ER þ ð1� gt þ y�1gtÞð1� a1EH1Þð1� 2ERÞ�ðy þ gc � 1Þg�1

c ga2EL1

�

= ff1�

� ðy þ gc � 1Þg�1c ð1� a2EL1Þ½ER þ ð1� 2ERÞð1� gt þ y�1gtÞ�

� ð1� gt þ y�1gtÞERgs1 � ðy þ gc � 1Þg�1c ð1� ERÞa2EL1ga1EH1

�ð29Þ

If the low and high pressure ratio compressors and the turbine are ideal (gc ¼ gt ¼ 1), theintercooling process is not involved in the cycle (x ¼ 1 and EI ¼ 0) and there is no pressure drop inthe cycle (D1 ¼ D2 ¼ 1), Eqs. (20) and (21) become [16]

P ¼ f1�

� yð1� a2EL1Þ½ER þ y�1ð1� 2ERÞ� � y�1ER � y�1ð1� ERÞa2EL1ga1EH1s1

þ f½1� y�1ð1� a1EH1ÞER� � y½ER þ y�1ð1� a1EH1Þð1� 2ERÞ � ð1� ERÞa1EH1�ga2EL1

�

= f1�

� y�1ð1� a1EH1ÞER � yð1� a2EL1Þ½ER þ y�1ð1� a1EH1Þð1� 2ERÞ�gCL

�ð30Þ

g ¼ 1� a2EL1y�1ð1�

� ERÞa1EH1s1 þ fy�1ð1� a1EH1ÞER þ ½yER þ ð1� a1EH1Þ� ð1� 2ERÞ� � 1ga2EL1

�= ff1�

� ð1� a2EL1Þ½yER þ ð1� 2ERÞ� � y�1ERgs1� yð1� ERÞa2EL1ga1EH1

�ð31Þ

If the thermal capacity rates of the heat reservoirs are infinite (CH ¼ CL ! 1), the low and highpressure ratio compressors and the turbine are ideal (gc ¼ gt ¼ 1), the intercooling process is notinvolved in the cycle (x ¼ 1 and EI ¼ 0) and there is no pressure drop in the cycle (D1 ¼ D2 ¼ 1),Eqs. (20) and (21) become [27]

P ¼ f1�

� ð1� ELÞ½yER þ ð1� 2ERÞ� � y�1ER � y�1ð1� ERÞELgEHs1 � fy�1ð1� EHÞER

þ ½yER þ ð1� EHÞð1� 2ERÞ þ ð1� ERÞEH� � 1gEL

�= 1�

� y�1ð1� EHÞER

� ð1� ELÞ½yER þ ð1� EHÞð1� 2ERÞ��

ð32Þ

g ¼ 1� ELy�1ð1� ERÞEHs1 þ fy�1ð1� EHÞER þ ½yER þ ð1� EHÞð1� 2ERÞ� � 1gEL

ff1� ð1� ELÞ½yER þ ð1� 2ERÞ� � y�1ERgs1 � ð1� ERÞELgEH

ð33Þ

If the regenerated and intercooling processes are not involved in the cycle (ER ¼ 0, x ¼ 1 andEI ¼ 0), Eqs. (20) and (21) become [14]

P ¼ f1�

� ðy þ gc � 1Þg�1c ð1� a2EL1Þð1� gt þ y�1DgtÞ � ð1� gt þ y�1DgtÞa2EL1ga1EH1s1

� fðy þ gc � 1Þg�1c ½ð1� gt þ y�1DgtÞð1� a1EH1Þ þ a1EH1� � 1ga2EL1

�

= 1�

� ðy þ g � 1Þg�1ð1� a E Þð1� g þ y�1Dg Þð1� a E Þ�

ð34Þ
c c 2 L1 t t 1 H1


g ¼ 1� ð1�

� gt þ y�1DgtÞa2EL1a1EH1s1 þ ½ð1� gt þ y�1DgtÞð1� a1EH1Þ

� ðy þ gc � 1Þg�1c � 1�a2EL1

�= f½1�

� ð1� gt þ y�1DgtÞðy þ gc � 1Þg�1c ð1� a2EL1Þ�s1

� ðy þ gc � 1Þg�1c a2EL1ga1EH1

�ð35Þ

If the regenerated and intercooling processes are not involved in the cycle (ER ¼ 0, x ¼ 1 andEI ¼ 0), the low and high pressure ratio compressors and the turbine are ideal (gc ¼ gt ¼ 1), thereis no pressure drop in the cycle (D1 ¼ D2 ¼ 1) and the thermal capacity rates of the heat reservoirsare infinite (CH ¼ CL ! 1), Eqs. (20) and (21) become [23,28]

P ¼ f½ð1� y�1Þs1 � y þ 1�EHELg=½1� ð1� ELÞð1� EHÞ� ð36Þ

g ¼ 1� ðy�1s1 � 1Þ=ðs1 � yÞ ð37Þ

If the heat transfers between the working fluid and the heat reservoirs can be executed ide-ally (UH ¼ UL ! 1), the low and high pressure ratio compressors and the turbine areideal (gc ¼ gt ¼ 1), there is no pressure drop in the whole cycle (D1 ¼ D2 ¼ 1) and the thermalcapacity rates of the heat reservoirs are infinite (CH ¼ CL ¼ CI ! 1), Eqs. (20) and (21) become[29,30]

P ¼ ð1þ y�1Þs1 � yð1� EIÞ þ ð1� xEIÞ � ðyx�1 � 1ÞEIs2 ð38Þ

g ¼ 1� ð1� ERÞy�1s1 þ yð1� EI1ÞER � ð1� xEIÞ þ ðyx�1ER � 1ÞEIs2ð1� y�1ERÞs1 � yð1� ERÞð1� EIÞ � yx�1ð1� ERÞEIs2

ð39Þ

Eqs. (38) and (39) are the results of conventional thermodynamic analysis.

3. Results and discussion

To see how the component (regenerator, intercooler and hot and cold side heat exchangers)effectivenesses, the compressor and turbine efficiencies, the pressure recovery coefficients, the heatreservoir inlet temperature ratio and the cooling fluid in the intercooler and the cold side heatreservoir inlet temperature ratio influence the dimensionless power (P ), the efficiency (g) and theintercooling pressure ratios corresponding to the optimal power or efficiency, detailed numericalexamples are provided.

The characteristics of the dimensionless power (P ) and the efficiency (g) versus total pressureratio and intercooling pressure ratio with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, Cwf ¼ 1:0kW/K, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00 are shown in Figs. 2 and 3. If the total pressureratio is fixed, there exists an optimal intercooling pressure ratio (ðp1ÞPopt

) that makes P reach theoptimum (P opt) and an optimal intercooling pressure ratio (ðp1Þgopt ) that makes g reach the op-timum (gopt), respectively. If the total pressure ratio is variable, there exists an optimal totalpressure ratio (pPmax

) and an optimal intercooling pressure ratio (ðp1ÞPmax) that make P reach the

maximum (Pmax) and an optimal total pressure ratio (pgmax) and an optimal intercooling pressure

ratio (ðp1Þgmax) that make g reach the maximum (gmax), respectively.

Fig. 3. Efficiency versus total pressure ratio and intercooling pressure ratio.

Fig. 2. Dimensionless power versus total pressure ratio and intercooling pressure ratio.


3.1. The total pressure ratio is fixed

In the calculation, k ¼ 1:4, p ¼ 12, Cwf ¼ 1:0 kW/K, s1 ¼ 4:33 and s2 ¼ 1:00 are set.Fig. 4 shows the effect of ER on P and g versus p1 with EH1 ¼ EL1 ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85

and D1 ¼ D2 ¼ 0:96. Fig. 5 shows the effect of EI1 on the characteristics of P and g versus p1 withEH1 ¼ EL1 ¼ ER ¼ 0:9, gc ¼ gt ¼ 0:85 and D1 ¼ D2 ¼ 0:96. Fig. 6 shows the effect of gc and gt

on the characteristics of P and g versus p1 with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9 and D1 ¼ D2 ¼ 0:96.Fig. 7 shows the effect of D1 and D2 on the characteristics of P and g versus p1 with EH1 ¼EL1 ¼ ER ¼ EI1 ¼ 0:9 and gc ¼ gt ¼ 0:85.

Figs. 4–7 indicate that P reaches, the optimal value (P opt) rapidly, then decreases gradually as p1

increases continuously within the range of (1;p), and P increases with increases in ER, EI1, gc, gt,D1 and D2. The efficiency g has similar characteristics.

3.2. The total pressure ratio is variable

Fig. 8 shows the effect of ER on the characteristics of P opt and gopt versus p withEH1 ¼ EL1 ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 9 shows the

Fig. 4. Dimensionless power (solid) and efficiency (dash) versus intercooling pressure ratio and effectiveness of the

regenerator.

Fig. 5. Dimensionless power (solid) and efficiency (dash) versus intercooling pressure ratio and effectiveness of the

intercooler.

Fig. 6. Dimensionless power (solid) and efficiency (dash) versus intercooling pressure ratio and compressor and turbine

efficiencies.


Fig. 7. Dimensionless power (solid) and efficiency (dash) versus intercooling pressure ratio and pressure recovery

coefficients.

Fig. 8. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and effectiveness of

the regenerator.

Fig. 9. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and effectiveness of

the intercooler.



effect of EI1 on the characteristics of P opt and gopt versus p with EH1 ¼ EL1 ¼ ER ¼ 0:9,gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 10 shows the effect of EH1 and EL1 onthe characteristics of P opt and gopt versus p with ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96,s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 11 shows the effect of gc and gt on the characteristics of P opt and gopt

versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 12 showsthe effect of D1 and D2 on the characteristics of P opt and gopt versus p with EH1 ¼EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 13 shows the effect of s1 on thecharacteristics of P opt and gopt versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85,D1 ¼ D2 ¼ 0:96 and s2 ¼ 1:00. Fig. 14 shows the effect of s2 on the characteristics of P opt and gopt

versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96 and s1 ¼ 4:33.Fig. 8 illustrates that when p is less than the critical value, P opt increases with the increase in ER,

but when p exceeds the critical value and goes on increasing, P opt decreases with the increase in ER.The optimal efficiency gopt has similar characteristics. The case that P opt and gopt decrease is aresult of the large total pressure ratio. When the total pressure ratio is so large that the outlettemperature at the high pressure ratio compressor is higher than the outlet temperature at theturbine, this leads to the working fluid in the regenerator not being heated, and instead results in a

Fig. 10. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and effective-

nesses of the hot and cold heat exchangers.

Fig. 11. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and compressor

and turbine efficiencies.

Fig. 12. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and pressure

recovery coefficients.

Fig. 13. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and heat reservoir

inlet temperature ratio.

Fig. 14. Optimal dimensionless power (solid) and optimal efficiency (dash) versus total pressure ratio and the cooling

fluid in the intercooler and the cold side heat reservoir inlet temperature ratio.


heat loss. Figs. 9–14 illustrate that P opt increases with increases in EI1, EH1, EL1, gc, gt, D1, D2 ands1, and decreases with an increase in s2. Figs. 8–14 indicate that the characteristic of P opt versus p


is a parabolic like curve. As p increases, P opt reaches its climax rapidly and then decreasesgradually. There exists an optimal total pressure ratio pPmax

that makes P opt reach the maximum,Pmax.

From Figs. 8–14, one can see that the characteristics of gopt versus p, p1, ER, EH1, EL1, EI1, gc, gt,D1 and D2 are similar to the characteristics of P opt versus p, p1, ER, EH1, EL1, EI1, gc, gt, D1 and D2.

Fig. 15 shows the effect of ER on the characteristics of ðp1ÞPoptand ðp1Þgopt versus p with

EH1 ¼ EL1 ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 16 shows the

effect of EI1 on the characteristics of ðp1ÞPoptand ðp1Þgopt versus p with EH1 ¼ EL1 ¼ ER ¼ 0:9,

gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 17 shows the effect of EH1 and EL1 onthe characteristics of ðp1ÞPopt

and ðp1Þgopt versus p with ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85,D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 18 shows the effect of gc and gt on the characteristicsof ðp1ÞPopt

and ðp1Þgopt versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, D1 ¼ D2 ¼ 0:96, s1 ¼ 4:33 ands2 ¼ 1:00. Fig. 19 shows the effect of D1 and D2 on the characteristics of ðp1ÞPopt

and ðp1Þgopt versusp with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, s1 ¼ 4:33 and s2 ¼ 1:00. Fig. 20 shows theeffect of s1 on the characteristics of ðp1ÞPopt

and ðp1Þgopt versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9,

Fig. 15. Intercooling pressure ratios at optimal dimensionless power (solid) and optimal efficiency (dash) versus total

pressure ratio and effectiveness of the regenerator.


pressure ratio and effectiveness of the intercooler.


pressure ratio and effectiveness of the hot and cold side heat exchangers.


pressure ratio and compressor and turbine efficiencies.


pressure ratio and pressure recovery coefficients.


gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96 and s2 ¼ 1:00. Fig. 21 shows the effect of s2 on the characteristicsof ðp1ÞPopt

and ðp1Þgopt versus p with EH1 ¼ EL1 ¼ ER ¼ EI1 ¼ 0:9, gc ¼ gt ¼ 0:85, D1 ¼ D2 ¼ 0:96and s1 ¼ 4:33.


pressure ratio and heat reservoir inlet temperature ratio.


pressure ratio and the cooling fluid in the intercooler and the cold side heat reservoir inlet temperature ratio.


Figs. 15–21 indicate that ðp1ÞPoptand ðp1Þgopt become larger and larger as p increases. The

intercooling pressure ratio corresponding to the optimum dimensionless power (ðp1ÞPopt) increases

with the increases in ER, EI1, EH1, EL1, gc, gt, D1, D2 and s2, and decreases with the increase in s1.The effects of EI1, gc, gt, D1, D2 and s1 on ðp1ÞPopt

are more slight than those of ER, EH1 and EL1 ands2., especially the pressure recovery coefficients (D1, D2) nearly do not, influence ðp1ÞPopt

. Theintercooling pressure ratio corresponding to the optimum efficiency (ðp1Þgopt ) increases with the

increase in ER, EH1, EL1 and s2, and decreases with the increases in EI1, gc, gt, D1 , D2 and s1. EI1, gc,gt, Cwf , D1, D2 and s1 do not affect ðp1Þgopt so evidently as ER and s2 do. Fig. 15 indicates that thecurves of ðp1ÞPopt

and ðp1Þgopt versus p overlap when ER ¼ 1 (the heat exchange process in theregenerator is fully ideal).

Figs. 22 and 23 show the characteristics of P opt versus gPoptand P gopt versus gopt, respectively.

The curve is found to be a loop shaped one, which is similar to that of a generalized irreversibleCarnot heat engine [31]. Pmax is slightly larger than the power corresponding to gmax, and gmax isalso slightly larger than the efficiency corresponding to Pmax. From Figs. 22 and 23, one can seethat there are two different efficiencies (or powers) corresponding to a power (or efficiency). So, if

Fig. 22. Optimal dimensionless power versus its corresponding efficiency and effectiveness of the intercooler.

Fig. 23. Optimal efficiency versus its corresponding dimensionless power and effectiveness of the intercooler.


the power (or efficiency) is given, one should choose the working state at which the efficiency (orpower) is larger.

4. Conclusion

In this paper, FTT is applied to study the effects of component (regenerator, intercooler, andhot and cold side heat exchangers) effectivenesses, the compressor and turbine efficiencies, thepressure recovery coefficients, the heat reservoir inlet temperature ratio and the cooling fluid inthe intercooler and the cold side heat reservoir inlet temperature ratio on the optimal power andthe corresponding intercooling pressure ratio, as well as the optimal efficiency and the corres-ponding intercooling pressure ratio of an irreversible closed Brayton cycle coupled to variabletemperature heat reservoirs. The analytical formulae of the dimensionless power and efficiency ofthe cycle, as functions of component (regenerator, intercooler and hot and cold side heat ex-changers) effectivenesses, the compressor and turbine efficiencies, the pressure recovery coeffi-cients, the heat reservoir inlet temperature ratio and the cooling fluid in the intercooler and thecold side heat reservoir inlet temperature ratio, are derived. The intercooling pressure ratio is


optimized for the optimal dimensionless power and the optimal efficiency, respectively. Themethod and the results are different from those in the classical analyses [32,33] in which theintercooling pressure ratios are generally determined according to the minimum specific workthe compressors consumed.

Acknowledgements

This paper is supported by The Foundation for the Author of National Excellent DoctoralDissertation of PR China (project no. 200136) and The National Key Basic Research andDevelopment Program of PR China (project no. G2000026301).

References

[1] Novikov II. The efficiency of atomic power stations (A review). At Energiya 1957;3(11):409.

[2] Chambadal P. Les centrales nuclearies. Paris: Armand Colin; 1957. pp. 41–58.

[3] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power output. Am J Phys 1975;43(1):22–4.

[4] Andresen B. Finite-time thermodynamics. Copenhagen: Physics Laboratory II, University of Copenhagen; 1983.

[5] De Vos A. Endoreversible thermodynamics of solar energy conversion. Oxford: Oxford University Press; 1992.

[6] Bejan A. Entropy generation minimization: The new thermodynamics of finite-size devices and finite-time

processes. J Appl Phys 1996;79(3):1191–218.

[7] Chen L, Wu C, Sun F. Finite time thermodynamic optimization of entropy generation minimization of energy

systems. J Non-Equilib Thermodyn 1999;24(4):327–59.

[8] Berry RS, Kazakov VA, Sieniutycz S, Szwast Z, Tsirlin AM. Thermodynamic optimization of finite time processes.

Chichester: Wiley; 1999. p. 471.

[9] Sieniutycz S, De Vos A, editors. Thermodynamics of energy conversion and transport. New York: Springer; 2000.

p. 335.

[10] Wu C, Chen L, Chen J, editors. Recent advances in finite time thermodynamics. New York: Nova Science; 1999.

p. 560.

[11] Chen L, Wu Y, Sun F. Finite time thermodynamic analysis for gas turbine cycle: Theory and application. Gas

Turbine Technol 2001;14(1):46–53 (in Chinese).

[12] Leff HS. Thermal efficiency at maximum work output: New results for old engines. Am J Phys 1987;55(7):602–10.

[13] Bejan A. Theory of heat transfer-irreversible power plants. Int J Heat Mass Transfer 1988;31(6):1211–9.

[14] Chen L, Sun F, Wu C. Performance analysis of an irreversible Brayton heat engine. J Instit Energy 1997;70(482):

2–8.

[15] Chen L, Sun F, Wu C, Kiang RL. Theoretical analysis of the performance of a regenerated closed Brayton cycle

with internal irreversibilities. Energy Convers Manage 1997;38(9):871–7.

[16] Chen L, Ni N, Cheng G, Sun F. FTT performance of a closed regenerated Brayton cycle coupled to variable-

temperature heat reservoirs. Proceedings of the International Conference On Marine Engineering, Shanghai, 4–8

November 1996. p. 3.7.1–7.

[17] Chen L, Ni N, Cheng G, Sun F, Wu C. Performance analysis for a real closed regenerated Brayton cycle via

methods of finite time thermodynamics. Int J Ambient Energy 1999;20(2):95–104.

[18] Chen L, Sun F, Wu C. Effect of heat resistance on the performance of closed gas turbine regenerative cycles. Int J

Power Energy Syst 1999;19(2):141–5.

[19] Chen L, Zheng J, Sun F, Wu C. Optimum distribution of heat exchanger inventory for power density optimization

of an endoreversible closed Brayton cycle. J Phys D: Appl Phys 2001;34(3):422–7.

[20] Chen L, Zheng J, Sun F, Wu C. Power density optimization for an irreversible closed Brayton cycle. Open Syst

Information Dyn 2001;8(3):241–60.


[21] Chen L, Zheng J, Sun F, Wu C. Power density optimization for an irreversible regenerated closed Brayton cycle.

Phys Scripta 2001;64(3):184–91.

[22] Chen L, Zheng J, Sun F, Wu C. Power density analysis and optimization of a regenerated closed variable-

temperature heat reservoir Brayton cycle. J Phys D: Appl Phys 2001;34(11):1727–39.

[23] Ibrahim OM, Klein SA, Mitchell JW. Optimum heat power cycles for specified boundary conditions. Trans ASME

J Engng Gas Turbine Power 1991;113(4):514–21.

[24] Roco JMM, Veleasco S, Medina A, Calvo Hernaudez A. Optimum performance of a regenerative Brayton thermal

cycle. J Appl Phys 1997;82(6):2735–41.

[25] Feidt M. Optimization of Brayton cycle engine in contact with fluid thermal capacities. Rev Gen Therm

1996;35(12):662–6.

[26] Cheng CY, Chen CK. Maximum power of an endoreversible intercooled Brayton cycle. Int J Energy Res

2000;24(6):485–94.

[27] Chen L, Sun F. The finite-time thermodynamics performance of a closed regenerated Brayton cycle with isentropic

compression and expansion processes. Gas Turbine Technol 1995;8(4):29–35 (in Chinese).

[28] Chen L, Sun F. The finite-time thermodynamics analysis of a closed Brayton cycle. Gas Turbine Technol

1994;(2):34–9 (in Chinese).

[29] Bejan A. Advanced engineering thermodynamics. New York: Wiley; 1988.

[30] Haywood RW. Analysis of engineering cycles. 3rd ed. Oxford: Pergoman Press; 1980.

[31] Chen L, Sun F, Wu C. Effect of heat transfer law on the performance of a generalized irreversible Carnot engine.

J Phys D: Appl Phys 1999;32(2):99–105.

[32] Vadasz P, Pugatsch J, Weiner D, On the optimal location and number of intercoolers in a real compression process.

ASME Paper No. 88-GT-44, 1988.

[33] Vadasz P, Weiner D. The optimal intercooling of compressor by a finite number of intercoolers. Trans ASME J

Energy Resour Technol 1992;114(2):255–60.

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