2nd Law Analysis of Brayton Rankine Cycle

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Second-law based thermodynamic analysis of

Brayton/Rankine combined power cycle

with reheat

A. Khaliqa,*, S.C. Kaushikb

aDepartment of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia,

New Delhi 110025, IndiabCentre for Energy Studies, Indian Institute of Technology, New Delhi 10016, India

Accepted 3 August 2003

Abstract

The aim of the present paper is to use the second-law approach for the thermodynamic

analysis of the reheat combined Brayton/Rankine power cycle. Expressions involving the

variables for specific power-output, thermal efficiency, exergy destruction in components of

the combined cycle, second-law efficiency of each process of the gas-turbine cycle, and second-

law efficiency of the steam power cycle have been derived. The standard approximation for air

with constant properties is used for simplicity. The effects of pressure ratio, cycle temperature-

ratio, number of reheats and cycle pressure-drop on the combined cycle performance para-

meters have been investigated. It is found that the exergy destruction in the combustion

chamber represents over 50% of the total exergy destruction in the overall cycle. The com-

bined cycle efficiency and its power output were maximized at an intermediate pressure-ratio,

and increased sharply up to two reheat-stages and more slowly thereafter.# 2003 Elsevier Ltd. All rights reserved.

1. Introduction

A development in the search for higher thermal-efficiency of conventional power

plant has been the introduction of combined-cycle plants. This is leading to the

development of gas turbines dedicated to combined-cycle applications, which has

been a subject of great interest in recent years, because of their relatively low initial

Applied Energy & (2004) & – &

www.elsevier.com/locate/apenergy

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

doi:10.1016/j.apenergy.2003.08.002

* Corresponding author.

E-mail addresses: [email protected].

http://www.elsevier.com/locate/apenergy/a4.3d

mailto:abd&underscore;[email protected]

mailto:abd&underscore;[email protected]

http://www.elsevier.com/locate/apenergy/a4.3d

http://www.sciencedirect.com/





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costs, and the short time needed for their construction. An optimum system for a

given power-generation duty may involve alternate cycle configurations, such as

compressor intercooling, turbine reheat, and steam injection into the gas turbine

combustor.

The early development of the gas/steam turbine plant, was described by Sieppel

and Bereuter [1]. Czermak and Wunsch [2] carried out the elementary thermo-

dynamic analysis for a practicable Brown Boveri 125 MW combined gas–steam

turbine power plant. Wunsch [3] claimed that the efficiencies of combined gas–steamplants were more influenced by the gas-turbine parameters like maximum tempera-

ture and pressure ratio than by those for the steam cycle and also reported that the

maximum combined-cycle efficiency was reached when the gas-turbine exhaust

temperature is higher than the one corresponding to the maximum gas-turbine effi-

ciency. Horlock [4], based on thermodynamic considerations, outlined more recent

developments and future prospects of combined-cycle power plants. Wu [5] describe

the use of intelligent computer software to obtain a sensitivity analysis for the com-

bined cycle. Cerri [6] analyzed the combined gas–steam plant, without reheat, from

the thermodynamic point of-view. In his analysis, he singled out the parameters that

most influence efficiency, and further reported that combined cycles exhibit a goodperformance if suitably designed, but if the highest gas-turbine temperatures are

used, expensive fuel must be utilized.

Nomenclature

C p Specific heat at constant pressure (kJ/kg K)e Specific exergy (kJ/kg)

h Specific enthalpy (kJ/kg)

n Number of reheat stages

p Pressure (kPa)

Q Heat per unit mass of fuel (KJ/kg)

R Gas constant (KJ/kg-K)

S

ge Entropy generation rate (W/K)

s Specific entropy (KJ/kg)

W Work per unit mass of gas (KJ/kg)

w Dimensionless specific exergy/work (w=e/C pT 0)AC Pressure ratio across the compressor

Ratio of specific heats

1,g First-law efficiency of gas-turbine cycle

1,Comb First-law efficiency of combined cycle

2,Comb Second-law efficiency of combined cycle

Maximum to minimum cycle temperature ratio ( =T 3/T 0)

hf Dimensionless heat-input (H f /C pT 0)

H f Heat input or enthalpy of reaction at standard condition (KJ/kg)

2 A. Khaliq, S.C. Kaushik / Applied Energy& (2004) & – &



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Reheat has been widely employed in aircraft engines. However, for industrial gas-

turbines, it is a technique that has only recently reached the stage of being con-

sidered a viable option for power augmentation. For a fixed overall pressure-ratio

and given power, the advantage of using reheat is that the turbine’s entry tempera-ture (TET) corresponding to the main combustor and reheater of the reheat cycle is

lower than the TET of a simple cycle. Hence, the costs related to the use of expensive

superalloys to withstand high temperatures could be reduced as described by Cunha

et al. [7]. There is a reduction in efficiency, since more fuel is injected at a lower

pressure so producing less power than that which would be obtained if all the fuel

were injected in the main combustor. In combined-cycle applications, the increased

amount of heat in the exhaust gas is not actually lost and it may improve the com-

bined-cycle characteristics. Andriani et al. [8] carried out the analysis of a gas tur-

bine with several stages of reheat for aeronautical applications. Polyzakis [9] carried

out the first-law analysis of reheat industrial gas-turbines use in a combined cycle

and suggested that the use of reheat is a good alternative for combined-cycle appli-

cations. But the performance analysis based on the first-law alone is inadequate and

a more meaningful evaluation must include a second-law analysis. One reason that

such an analysis has not gained much engineering use may be the additional com-

plication of having to deal with the ‘‘combustion irreversibility’’, which introduced

an added dimension to the analysis. Second-law analysis indicates the association of

exergy destruction with combustion and heat-transfer processes and allows a ther-

modynamic evaluation of energy conservation in thermal power cycles.

It became apparent to the current authors that, although there was sufficient lit-erature on combined power-cycle with reheat, no systematic second-law analysis of

these cycles has been reported. The objective of the present paper is to develop a

systematic and improved second-law based thermodynamic methodology for the

analysis of reheat combined gas–steam power plant.

2. System description

A schematic diagram of a combined Brayton/Rankine power cycle with reheat is shown

in Fig. 1. The gas turbine is shown as a topping plant, which forms the high-tempera-

ture loop, whereas the steam plant forms the low-temperature loop. The connectinglink between the two cycles is the heat-recovery steam generator (HRSG) working on

the exhaust of the gas turbine. A gas-turbine cycle consists of an air compressor (AC), a

combustion chamber (CC) and a reheat gas-turbine (RGT). The turbine’s exhaust-gas

goes to a heat-recovery steam-generator to generate superheated steam. That steam is

used in a standard steam power-cycle, which consists of a turbine (ST), a condenser

(C) and a pump (P). Both the gas and steam turbines drive electric generators.

3. Thermodynamic analysis

For the system operations in a steady state, the general exergy-balance equation is

given by Bejan [10]

A. Khaliq, S.C. Kaushik / Applied Energy& (2004) & – & 3





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W g ¼ W RGT W AC ð6Þ

The first law efficiency of the gas-turbine cycle is given by

1;g ¼ W g

DH f ð7Þ

The total exergy of the fuel input for the gas turbine cycle with reheat is

ef ¼ ef ;CC þ eRGT ð8Þ

If we define maximum to minimum cycle temperature ratio as ¼ T max

T min¼

T 3

T 0

,

then the exergy associated with the fuel can be expressed as

ef ¼ CarnotDH f ¼ 1 1

DH f ð9Þ

The second-law efficiency of the gas-turbine cycle may be defined as

2;g ¼ W g

ef ð10Þ

The gas stream leaving the turbine at state 4 enters the steam power-cycle, where a

fraction 2,ST of its exergy (e4) is recovered as shaft work and the remaining exergy

destroyed by irreversibilities.

W ST ¼ 2;STe4 ð11Þ

Dividing by C pT 0, Eq. (11) becomes

W ST ¼ 2;STW 4 ð12Þ

The first-law efficiency of the combined power cycle is given by

1;Comb ¼ W g þ W ST

DH f ð13Þ

The gas-turbine’s specific work-output with single-stage reheat, on the basis of the

same expansion ratio and efficiency of each turbine and full reheat, and assuming air

as a perfect gas, may be given asW g ¼ C p 2 h3 h4ð Þ h2 h1ð Þ½ ð14Þ

where AC and RGT are the adiabatic efficiencies of the compressor and turbine.

For a system with ‘n’ stages of reheat, we would have

W g ¼ C p n þ 1ð ÞRGTT max 1 1

RGT

T 1

AC 1

=AC

ð16Þ

Dividing by C pT 0, the dimensionless specific power-output becomes

wg ¼ n þ 1ð ÞRGT RGT AC ACð Þ ð17Þ

where AC=AC 1 and RGT ¼ 1 1

RGT




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The fuel input or heat input (H f or Qin) per unit mass of the cycle for the single

stage with full reheat is given by

Qin ¼ DH f ;CC þ DH f ;RGT

ð18ÞFor a perfect gas, it may be expressed as

DH f ¼ Qin ¼ C P T max T 1 T 2

AC AC þ RGTT max RGT

ð19Þ

For ‘n’ reheat stage, it becomes

Qin ¼ C P T max T 1 T 2 AC

ACþ nRGTT max RGT

ð20Þ

Dividing by C pT 0, Eq. (20) may be written as

qin ¼ 1 AC=AC þ RGT RGT ð21Þ

Using Eqs. (17) and (21), the first-law efficiency of the gas-turbine cycle becomes

1;g ¼ W g

qin¼

n þ 1ð ÞRGT RGT AC=AC

1 AC=AC þ nRGT RGTð22Þ

Using Eqs. (12), (13), (17) and (21), the first law efficiency of the combined cycle

may be expressed as

1; comb ¼

n þ 1ð ÞRGT RGT AC=2;STw4 1 AC=AC þ nRGT RGT½ ð23Þ

This shows that the first-law efficiency of the combined cycle is a function of

temperature ratio ‘ ’, compressor’s pressure-ratio ‘ AC’, number of reheat stages ‘n’

and the pressure drop in the heat-transfer devices.

The second-law efficiency of combined cycle may be defined as

2; comb ¼ W g þ W ST

ef ¼

1;Comb

Carnotð24Þ

Using Eqs. (23) and (9) in Eq. (24),

2; comb ¼n þ 1ð ÞRGT RGT AC=AC þ 2;STw4

1 AC=AC þ nRGT RGT½ 1ð Þ

ð25Þ

where w4= 41-ln 4.

4. Relation between compressor and turbine pressure-ratios

The turbine expansion ratio RGT may be expressed in terms of the compressionratio and the pressure drop in each of the heat-transfer devices, involved. If pin, and

pout are the inlet and outlet pressures for each heat-transfer device, then




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pout ¼ pin ð26Þ

where =1 pin pout pin

¼ 1 D p p

The quantity p/p is known as the relative pressure-drop and b may be called thepressure-drop factor.

If cc is the pressure-drop factor or percentage pressure-drop in the combustion

chamber, R in reheater and g in heat recovery steam-generator, then

p3 ¼ CC p2 ð27Þ

pRo ¼ pRiR ð28Þ

pgo ¼ g pgi ð29Þ

Combining Eqs. (27)–(29), we have

p3

pRi

pRo

pgi

¼ CCRg

p2

pgo

¼ CCRgAC ð30Þ

Now

p3

pRi ¼

pRo

pgi ¼ AC

For a system with one stage of reheat,

RGTð Þ2¼ CCRgAC ð31Þ

RGT ¼ CCRgAC 1=2

ð32Þ

For two reheat-stages,

RGT ¼ CC2RgAC

1=3ð33Þ

For n reheat-stages,

RGT ¼ CCnRgAC

1= nþ1ð Þð34Þ

The traditional first-law efficiency of a steam turbine cycle is

1;ST ¼ W STQST

ð35Þ




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Its second-law efficiency has been defined by Eq. (11). Thus the ratio of its second-

law efficiency to its first law efficiency is just the ratio of the heat supplied to the

HRSG per unit mass of hot gas to the specific exergy of the hot gas entering the

HRSG. If the gas with a constant specific heat, enters the boiler at T 4 and leaves atT ex, then

2;ST

1;ST

¼ W ST

e4

QST

W ST¼

QST

e4

ð36Þ

For a constant-pressure process, by dividing by C pT 0

2;ST

1;ST

¼ 4 ex

4 1 ‘n 4ð37Þ

This is computed in Table (9) versus 4 with ex as the variable parameter. The

second-law efficiency of the steam-turbine cycle is larger than the first-law efficiency

so long as 4<1+1n 4, a condition satisfied in any practical steam-turbine bottom-

ing cycle.

For the purpose of combined cycle efficiency computations presented based on

Eqs. (23) and (25), the second-law efficiency of the steam-turbine cycle was assumed

to follow the trend shown in Fig. A1, which was plotted using the correlation

developed in the Appendix. The second-law efficiency (2,ST) is zero for 4<2, where

the steam-turbine cycle was judged impractical linearly from 48% at 4=2 to 70%

at 4=3.25, and constant at 70% for 5> 4>3.25.

5. Evaluation of component’s exergy destructions

5.1. Compressor (AC)

The second-law efficiency of a compression process (1–2) is the ratio of the

increased exergy to the work input: thus

2;AC ¼ e2 e1

W ACð38Þ

For frictionless reversible adiabatic or isothermal compressions, no entropy is

generated or exergy destroyed and 2,AC=1. In a real compressor of adiabatic effi-

ciency AC, for an infinitesimal adiabatic increase in pressure d p, the temperature

increase dT is greater than the isentropic value dT s.

dT ¼ dTs

ACð39Þ

For a perfect gas using the isentropic relation, we have

dTsT

¼ -1

dp p

ð40Þ




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The canonical relation is

ds ¼ C pdTs

T

Rd p

p

ð41Þ

Using Eqs. (39) and (41), the entropy generated during the compression process is

dsgen ¼ C pdTs

AC R

d p

p ð42Þ

Using Eqs. (40) and (42), we have

dsgen ¼ 1 AC

AC

Rd p

p ð43Þ

The exergy destroyed is obtained after multiplying Eq. (43) by T 0 and then inte-

grating. After non-dimensionalizing by dividing with C pT 0, the dimensionless exergy

destruction may be given as

wD;AC ¼ 1 AC

AC

‘nr ð44Þ

Eq. (44) accounts for the exergy destroyed within the compressor. The compressor

work for the infinitesimal adiabatic stage is C pdT . After using Eqs. (39) and (40) for

a perfect gas, the compressor work in dimensionless form may be given as

wAC ¼

1

ACð 2

1

T

T 0

d p

p ð45Þ

Unlike the exergy destroyed, this depends on the local temperature. The com-

pression work for adiabatic compression may be obtained by using Eqs. (40) and

(45) as

wAC;ad ¼ r1=AC 1 ð46Þ

Applying the exergy balance and using Eq. (38), the corresponding second-law

efficiency for the adiabatic compression process may be given by

2;ACad ¼ 1

1 AC

AC

‘nr

r1=AC 1 ð47Þ

5.2. Combustion chamber (CC)

The heat addition in the combustion chamber (H f,CC) may be defined as

DH f ;CC ¼ Q

m ¼ C p T 3 T 2ð Þ ð48Þ

After dividing Eq. (48) by C pT 0, it may be expressed as




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Dhf ;CC ¼ 2 ð49Þ

The exergy associated with Hf,CC is

ef ;CC ¼ DH f ;CC 1 1= ð Þ ð50Þ

Using Eq. (49) and dividing Eq. (50) by C pT 0, the dimensionless exergy associated

with fuel may be obtained as

wf ;CC ¼ 1ð Þ 2ð Þ

ð51Þ

The increase in exergy per unit mass of fuel is given by

e3 e2 ¼ h3 h2ð Þ T 0 s3 s2ð Þ ð52Þ

After dividing by C pT 0 and using Eqs. (41), (49) and (26), it may further be

expressed as

w3 w2 ¼ 2 ‘n

2þ ‘nCC ð53Þ

The dimensionless exergy destruction (wD,CC) in the combustion chamber can be

expressed using Eqs. (3) and (53), as

wD;CC ¼ 2

þ ‘n

2 ‘nCC 1 ð54Þ

The second-law efficiency for the combustion chamber is the ratio of the increased

exergy over the exergy input and is given by

2;CC ¼ w3 w2

wf ;CC

ð55Þ

Using Eqs. (49) and (53),

2;CC ¼

2 ‘n

2þ ‘nCC

2

ð Þ 1ð Þ ð56Þ

This shows that the second-law efficiency of the combustion chamber depends on

the compressor’s discharge temperature, pressure-drop in the combustion chamber

and the maximum cycle temperature.

5.2.1. Reheat gas-turbine (RGT)

For an adiabatic expansion in a turbine with an adiabatic efficiency RGT, the

temperature-drop dT for a pressure drop dp is smaller than the corresponding isen-

tropic value dT s.

RGT ¼ dT

dTs ð57Þ




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Using Eqs. (40), (41) and (43), we see that the entropy generated in the adiabatic

stage is

dsgen ¼ 1 RGTð ÞRd p

p ð58Þ

By multiplying Eq. (58) by T 0 and integrating, and, thereafter dividing by C pT 0,

the exergy destruction may be obtained as

wD;RGT ¼ 1 RGTð Þ‘nr ð59Þ

It accounts only for exergy destroyed within the turbine but not for reheat pres-

sure-losses or heat-transfer losses. The expansion work C pdT , after using Eqs. (57)

and (40), may be expressed as

wRGT ¼ RGT 1ð Þ

ð 43

T T 0

dp p

ð60Þ

and depends on the pressure–temperature path. For the adiabatic expansion starting

at T 3 after integrating Eq. (60) and using ( =T 3/T 0), it may also be expressed as

wRGT;ad ¼ 1 rRGTð Þ ð61Þ

The second-law efficiency of the expansion process is the ratio of work output

over decrease in the exergy of the gas, and is given by

2;RGT ¼ wRGT

w3 w2 þ wRGT

ð62Þ

Using Eqs. (61), (53) and (62), the second-law efficiency of the expansion process

in the gas turbine cycle may be given as

2;RGTad ¼ 1 þ

1 RGTð Þr‘nr

1 rRGTð Þ

1

ð63Þ

This shows that the second-law efficiency of the reheat gas-turbine increases with y

since a larger proportion of the available work lost at higher temperatures may berecovered.

6. Optimum pressure-ratio

The optimum pressure ratio for maximum work output of a gas turbine, taking

into account the adiabatic efficiencies of the compressor and turbine, can be

obtained by differentiating Eq. (17), w.r.t. pAC as

@wg

@AC¼ 0 ð64Þ




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This gives

ACð Þopt¼ 1

ACRGT

2 1 ð Þ

ð65Þ

This shows that the optimum pressure-ratio depends on the adiabatic efficiencies

of the turbine and compressor, as well as the cycles temperature-ratio.

7. Numerical results and discussion

Based upon the methodology developed and the equations derived here, the

combined-cycle efficiency, exergy destruction as well as the second-law efficiency of

each process have been evaluated.

For the results, we made the following assumptions; adiabatic efficiencies of

compressor and gas turbine are 0.9 and 0.85, respectively; pressure drops in the

primary combustor are 3%, in each reheater 2% and in the HRSG 4%. The gas is

assumed to have constant properties with =1.4, R=287 J/kg K. For illustration of

the results, the pressure ratio was taken as 32, cycle-temperature ratio as 5, two

reheats and no intercooling.

Table 1 shows the variation of performance parameters of the compressor and gas

turbine with the pressure ratio. The second-law efficiency of the adiabatic com-pressor increases with pressure ratio because the absolute values of the work input

and exergy increase are both larger and the magnitude of exergy destruction in the

adiabatic compressor increases with the increase in pressure ratio.

It is also seen from Table 1 that, the first-law efficiency of the adiabatic turbine

increases with the increase in pressure ratio. The second-law efficiency decreases with

the pressure ratio, but increases with the cycle temperature ratio since a greater

proportion of the available work lost at the higher temperature may be recovered.

The exergy destruction in the reheat turbine increases with the pressure ratio, the

number of reheat stages and the pressure drop in each reheater as shown in Table 2.

Table 3(a) and (b) show that the first-law and second-law efficiencies of the com-bined cycle increases up to the pressure ratio of 32, then they start decreasing with

increases in the pressure ratio. But it is interesting to note that the second-law effi-

ciency of the combined cycle is greater than the first-law efficiency for same pressure-

ratio.

Table 4 shows that if the pressure ratio is too low, then the gas-turbine cycle and

combined-cycle efficiencies and their specific work-outputs drop, whereas the steam

cycle work-output increases due to the high gas-turbine exhaust temperature T 4. At

an intermediate pressure-ratio, both the efficiency and specific work peak. If the

pressure ratio is too high, the compressor and turbine works increase but their dif-

ference, the net gas-turbine work output drops. The absolute magnitude of exergydestroyed in both compressor and turbine increases as the logarithm of pressure

ratio. The exergy lost in the reheat turbine also increases due to the lower mean




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temperature of reheat. The steam-turbine cycle output suffers with the lower

exhaust-gas temperature. The second-law efficiency of each cycle is greater than the

first-law efficiency for the given operating parameters.

It is seen from Table 5 that the exergy destruction in the combustion chamber

decreases with the pressure ratio, but increases with the cycle temperature ratio y,

and the second-law efficiency of the primary combustor behaves in reverse as isknown from the second-law analysis.

The exergy destructions due to heat-transfer irreversibility (HRSG), condenser-

heat rejection, irreversibilities of the steam turbine and pump, and the first-law effi-

ciency of the steam turbine cycle increase with an increase in the gas-turbine’s

exhaust temperature, but the second-law efficiency declines with an increase in the

exhaust-gas’s temperature above the minimum temperature that can operate the

steam cycle. This minimum gas temperature is constrained by the required superheat

steam and or the pinch point on the HRSG as shown in Table 8.

Table 6 shows that increasing the maximum cycle temperature gives a significant

improvement in both efficiency and specific work-output. The gas-turbines cycleefficiency drops, but its net specific work-output increases with the number of reheat

stages. Both efficiency and specific work increase with the increase in number of

Table 1

Effect of pressure ratio on the performance of compressor and gas turbine

AC AC 1,AC wD,AC 2,AC RGT

RGT 1,RGT wD,RGT 2,RGT

1 1.000 0.900 0.000 0.900 0.950 0.985 0.850 0.000 0.995

2 1.219 0.890 0.022 0.910 1.151 1.041 0.862 0.029 0.955

4 1.485 0.880 0.043 0.920 1.369 1.094 0.872 0.059 0.941

8 1.811 0.870 0.066 0.929 1.628 1.149 0.883 0.089 0.924

16 2.208 0.860 0.088 0.937 1.76 1.175 0.894 0.118 0.903

32 2.691 0.850 0.110 0.945 2.303 1.269 0.904 0.148 0.876

64 3.281 0.832 0.132 0.951 2.739 1.333 0.912 0.178 0.844

128 3.999 0.820 0.154 0.957 3.257 1.401 0.920 0.207 0.806

Table 2

Effect of number of reheat stages (n) and pressure drops in the reheater (R) on the exergy destruction in

the reheat gas-turbine

Number of

reheat stages (n)

wD,RGT

R=1.0 R=0.98 R=0.96

0 0.1485 0.1485 0.1485

1 0.2257 0.2331 0.2380

2 0.2203 0.2277 0.2389

3 0.2107 0.2223 0.2407

4 0.2023 0.2201 0.2441

5 0.1954 0.2177 0.24976 0.1908 0.2166 0.2565

7 0.1858 0.2173 0.2643




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reheat stages for the steam cycle which benefits from a higher gas-temperature. The

combined cycle efficiency and specific work-output increase sharply in going from

one to two reheats and more slowly thereafter, It was interesting to note that thespecific power increases by a factor of 2.5 for the two reheats as shown in Table 7.

This may well justify the additional capital cost of the reheat system.

Table 9 shows that the second-law efficiency of steam-turbine cycle is larger than

the first-law efficiency so long as <1+ln 4, a condition satisfied in any practical

steam-bottoming cycle. It is shown that the second-law efficiency of a given steam

cycle declines with increasing gas-temperature above the minimum that can operate

this cycle. This minimum gas-temperature is constrained by the required steam

superheat and/or the ‘‘pinch point’’ on the heat exchanger.

Fig. 2 shows the effect of increasing the pressure ratio and the cycle-temperature

ratio on the first-law efficiency of the gas-turbine cycle. The increase in pressure ratioincreases the overall thermal efficiency at a given maximum temperature. However

increasing the pressure ratio beyond a certain value at any given maximum

Table 3

(a) Effect of pressure ratio (AC) and cycle temperature ratio ( ) on the first-law efficiency of the combined

cycle for two stages of reheat. (b) Effects of pressure ratio (pAC) and cycle temperature ratio ( ) on the

second-law efficiency of the combined cycle for two stages of reheat

=4 =4.5 =5 =5.5 =6

AC RGT 4 1,Comb 4 1,Comb 4 1,Comb 4 1,Comb 4 1,Comb

(a)

8 1.920 3.413 51.1 3.840 53.70 4.267 55.80 4.69 57.5 5.0 59.0

16 2.42 3.227 52.3 3.630 55.76 4.034 58.20 4.438 60.0 4.84 61.57

32 3.048 3.050 50.7 3.433 56.36 3.814 59.20 4.195 61.45 4.57 63.23

64 3.840 2.880 46.7 3.245 54.80 3.606 58.80 3.960 61.65 4.12 63.80

128 4.838 2.727 38.0 3.068 49.00 3.409 56.50 3.750 60.30 3.72 63.20

(b)8 1.920 3.413 68.13 3.840 69.11 4.267 69.75 4.69 70.30 5.0 70.80

16 2.42 3.227 69.73 3.630 71.76 4.034 72.75 4.438 73.35 4.84 73.80

32 3.048 3.050 67.6 3.433 72.53 3.814 74.0 4.195 75.12 4.57 75.80

64 3.840 2.880 62.26 3.245 70.52 3.606 73.5 3.960 75.36 4.12 76.56

128 4.838 2.727 50.66 3.068 63.06 3.409 70.62 3.750 73.71 3.72 75.84

Table 4

Effect of pressure ratio on the first-law and second-law efficiencies of various cycles

AC 1,g 2,g 1,ST 2,ST 1,Comb 2,Comb Carnot

8 27.8 34.75 28.00 43.82 55.85 69.81 80.00

16 33.0 41.25 25.17 40.18 58.17 72.71 80.00

32 35.95 44.93 23.25 37.82 59.2 74.00 80.00

64 36.7 45.87 22.40 37.1,6 58.8 73.50 80.00

128 34.4 43.00 22.00 37.00 56.40 70.50 80.00




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temperature can actually result in lowering the gas-turbine’s cycle efficiency. It

should also be noted that the very high-pressure ratios tend to reduce the operating

range of the compressor.

Fig. 3 shows that the maximum work per kilogramme of air occurs at a muchlower pressure-ratio than the point of maximum efficiency for the same maximum

temperature.

Table 5

Effect of pressure ratio (AC) and cycle temperature ratio ( ) on exergy destruction and second law effi-

ciency of combustion chamber (CC) for two reheats

AC =4 =4.5 =5 =5.5 =6

wD,CC 2,CC wD,CC 2,CC wD,CC 2,CC wD,CC 2,CC wD,CC 2,CC

1 1.627 0.133 1.717 0.126 1.800 0.119 1.877 0.112 1.949 0.1066

2 1.484 0.163 1.568 1.550 1.647 0.146 1.719 0.138 1.423 0.1317

4 1.353 0.192 1.430 0.186 1.502 0.176 1.570 0.167 1.634 0.1590

8 1.236 0.214 1.303 0.210 1.368 0.204 1.439 0.196 1.490 0.1880

16 1.137 0.215 1.190 0.222 1.248 0.223 1.305 0.219 1.359 0.2140

32 1.060 0.176 1.103 0.207 1.148 0.222 1.195 0.228 1.241 0.2300

64 1.010 0.058 1.036 0.134 1.068 0.180 1.103 0.206 1.141 0.2210

128 0.990 0.028 0.997 0.100 1.014 0.061 1.036 0.126 1.062 0.1680

Table 6

Effect of cycle temperature-ratio on efficiencies of various cycles

Temperature

ratio ‘ ’

Z1,g 2,g 1,ST 2,ST 1,Comb 2,Comb Carnot

4 30.30 40.40 20.50 35.60 50.80 67.70 75.00

4.5 33.70 43.33 22.57 37.90 56.27 72.40 77.70

5.0 36.00 45.00 23.30 37.95 59.30 74.12 80.00

5.5 37.60 46.95 23.90 37.86 61.50 75.18 81.80

6.0 38.85 46.62 24.39 37.63 63.24 75.89 83.336.5 39.80 47.05 24.86 37.50 64.66 76.40 84.60

Table 7

Effects of number of reheat stages (n) on work output and efficiencies of various cycles

n 1,g 1,ST 1,Comb W g wg+w4 wComb qin Carnotqin

0 43.50 9.00 52.50 0.950 1.500 1.20 2.100 1.700

1 37.28 66.80 57.90 1.403 2.513 2.18 3.762 3.00

2 36.7 69.57 59.77 1.644 3.109 2.67 4.469 3.5753 36.2 70.63 60.33 1.76 3.425 2.926 4.85 3.880

4 35.9 71.40 60.75 1.828 3.63 3.089 5.086 4.068

5 35.7 71.90 61.00 1.865 3.756 3.186 5.224 4.180




ARTICLE IN PRESS

Thus, a cursory inspection of the efficiency indicates that the gas-turbine cycleefficiency can be improved by increasing the pressure ratio, or increasing the tur-

bine’s inlet-temperature.

8. Conclusion

An improved second-law analysis of the combined power-cycle with reheat has

shown the importance of the parameters examined. The analysis has included the

exergy destruction in the components of the cycle and an assessment of the effects of

pressure ratio, temperature ratio and number of reheat stages on the cycle perfor-mance. The exergy balance or second-law approach presented facilitates the design

and optimization of complex cycles by pinpointing and quantifying the losses. By

Table 8

Exergy destruction as a percentage of heat added, in the components of the steam-turbine cycle: T 0=291

K, T ex=420 K, condenser pressure=0.045 bar (304 K), steam-turbine efficiency 90%, pump efficiency

70%

Exhaust-gas

temperature ratio

Exhaust

availability

Heat-transfer

irreversibility

Condenser loss

and rejection

Irreversibility

of turbine and

pump

Steam cycle

work output

2.00 73 13 6 4 49

2.25 81 18 5 6 52

2.50 85 16 6 5 58

2.75 88 17 5 6 61

3.00 90 16 5 7 63

3.25 91 13 4 8 65

Table 9

Effects of gas temperature ratio 4 and exhaust temperature ratio ex on the ratio of efficiencies of the

steam cycle

4 ex=1 ex=1.5 ex=2.0 ex=2.5 ex=3.0 ex=3.5 ex=40

2;ST

1;ST

2;ST

1;ST

2;ST

1;ST

2;ST

1;ST

2;ST

1;ST

2;ST

1;ST

2;ST

1;ST

2 3.258 1.629 1.109 0.5540 – – –

3 2.218 1.664 1.109 0.5540 – – –

4 1.859 1.459 1.239 0.9290 0.6196 0.3098 – 5 1.673 1.464 1.255 1.0457 0.8366 0.6274 0.4180

6 1.558 1.4026 1.246 1.0909 0.9350 0.7792 0.6230

7 1.479 1.356 1.233 1.1099 0.9866 0.8633 0.7400

8 – – – – 1.0160 0.9145 0.8129

9 – – – – 1.0339 0.9478 0.8616

10 – – – – 1.0450 0.9705 0.8958

11 – – – – 1.0523 0.9865 0.9207




ARTICLE IN PRESS

placing reheat in the expansion process, significant increases in specific power output

and efficiency were obtained. The gains are substantial for one and two reheats, but

progressively smaller for subsequent stages. It is interesting to note that specific

power output (per unit gas flow) increases by a factor of 2.5 for the two reheats. This

may well justify the additional capital cost of the reheat system. Reheating byincreasing the specific power-output reduces the sensitivity of the cycle to component

losses.

Fig. 2. Effect of pressure ratio and turbines inlet temperature on the first-law efficiency of the gas–turbine

cycle.

Fig. 3. Pressure ratio for maximum work per kg of air.




ARTICLE IN PRESS

Appendix. Correlation for the second-law efficiency of the steam cycle

For a simple steam-cycle, the maximum second-law efficiency can be correlated

with the gas temperature T 4 for a fixed exhaust-gas temperature T ex.To find this correlation, calculations were done for several values of the tempera-

ture T 4. In each case, the steam-turbine cycle pressure and peak temperature T 5,ST

were first determined by setting the pinch point (saturation) and maximum steam-

temperatures at 5 and 20 K below the corresponding gas-temperature profile. Thus

the percentage of gas and steam enthalpies above the pinch point must be the same,

giving

T 4 T sat þ 50

T 4 T ex

¼ h5;st hsat;liq

h5;st h8;liq

ðA1Þ

which may be solved iteratively for the steam-turbine cycle pressure. In the following

calculations, the assumptions are:

Fig. A1. Second-law efficiency correlation for bottoming cycle.




ARTICLE IN PRESS

1. Ambient temperature T 0=291 K

2. Exhaust temperature T ex=420 K

3. Condenser pressure 0.045 bar (304 K)

4. Steam turbine and feed water pump have efficiencies 90 and 70% respectively.5. Saturation temperature (T sat)=T ex22 C.

For each T 4, these assumptions were applied, the pressure was found using Eq.

(A1) and the second-law efficiency (2,ST) is computed and is shown in Fig. A1,

which also shows the steam conditions and efficiency computed for each point.

References

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operating experience paper no. 82 GT-323]. ASME; 1982.

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[4] Horlock JH. Combined power plants—past, present, and future. Trans ASME, Journal of Engng for

Gas Turbine and Power 1995;117:608–16.

[5] Wu C. Intelligent computer-aided sensitivity analysis of multi-stage Brayton/Rankine combined

cycle. Energy Conv Mgmt 1999;40:215–32.

[6] Cerri G. Parametric analysis of combined gas–steam cycles. Trans, ASME J Engng Gas Turbine

Power 1987;109:46–54.

[7] Cunhas Alves MA, da Franca Mendes Carniero HF, de Barbosa JR, Travieso LE, Pildis P. An

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[8] Andriani R, Ghezzi U, Anntoni LFG. Jet engines with heat addition during expansion: a perfor-

mance analysis paper 99-0744]. AIAA; 1999.

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2nd Law Analysis of Brayton Rankine Cycle

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