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Rankine combined power cycle with reheat
Applied Energy 78 (2004) 179–197 www.elsevier.com/locate/apenergy
Second-law based thermodynamic analysis of Brayton/Rankine combined power cycle with reheat A. Khaliqa,*, S.C. Kaushikb a
Department of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi 110025, India b Centre 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 secondlaw 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 temperatureratio, number of reheats and cycle pressure-drop on the combined cycle performance parameters 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 combined 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 * Corresponding author. E-mail addresses: [email protected]. 0306-2619/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2003.08.002
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Nomenclature Cp e h n p Q R Sge s W w AC 1,g 1,Comb 2,Comb hf Hf
Specific heat at constant pressure (kJ/kg K) Specific exergy (kJ/kg) Specific enthalpy (kJ/kg) Number of reheat stages Pressure (kPa) Heat per unit mass of fuel (KJ/kg) Gas constant (KJ/kg-K) Entropy generation rate (W/K) Specific entropy (KJ/kg) Work per unit mass of gas (KJ/kg) Dimensionless specific exergy/work (w=e/CpT0) Pressure ratio across the compressor Ratio of specific heats First-law efficiency of gas-turbine cycle First-law efficiency of combined cycle Second-law efficiency of combined cycle Maximum to minimum cycle temperature ratio (=T3/T0) Dimensionless heat-input (Hf/CpT0) Heat input or enthalpy of reaction at standard condition (KJ/kg)
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 thermodynamic analysis for a practicable Brown Boveri 125 MW combined gas–steam turbine power plant. Wunsch [3] claimed that the efficiencies of combined gas–steam plants were more influenced by the gas-turbine parameters like maximum temperature 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 efficiency. 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 combined 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 good performance if suitably designed, but if the highest gas-turbine temperatures are used, expensive fuel must be utilized.
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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 considered 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 temperature (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 combined-cycle characteristics. Andriani et al. [8] carried out the analysis of a gas turbine 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 applications. 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 complication 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 thermodynamic evaluation of energy conservation in thermal power cycles. It became apparent to the current authors that, although there was sufficient literature 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-temperature loop, whereas the steam plant forms the low-temperature loop. The connecting link 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]
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Fig. 1. Schematic diagram of the combined Brayton/Rankine power cycle with reheat.
EW ¼
n X X X EQ þ me me ED i
i¼1
ð1Þ
out
in
for a single-stream flow
EW ¼ EQ þ mein meout meD
ð2Þ
After making exergy balances using Eq. (2) for the compressor, reheat turbine and combustion chamber, the following expressions can be obtained WAC ¼ ðe2 e1 Þ þ eD;AC
ð3Þ
ef;CC ¼ ðe3 e2 Þ þ eD;CC
ð4Þ
eRGT ¼ ðe4 e3 Þ þ WRGT þ eD;RGT
ð5Þ
where WAC and WRGT are the work done per unit mass for the compressor and reheat gas-turbine respectively and ‘e’ is the specific exergy The net work-output of the gas-turbine cycle is
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Wg ¼ WRGT WAC
ð6Þ
The first law efficiency of the gas-turbine cycle is given by 1;g ¼
Wg DHf
ð7Þ
The total exergy of the fuel input for the gas turbine cycle with reheat is ef ¼ ef;CC þ eRGT
If we define maximum to minimum cycle temperature ratio as then the exergy associated with the fuel can be expressed as 1 ef ¼ Carnot DHf ¼ 1 DHf
ð8Þ Tmax T3 ¼ , ¼ Tmin T0 ð9Þ
The second-law efficiency of the gas-turbine cycle may be defined as 2;g ¼
Wg 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. WST ¼ 2;ST e4
ð11Þ
Dividing by CpT0, Eq. (11) becomes WST ¼ 2;ST W4
ð12Þ
The first-law efficiency of the combined power cycle is given by 1;Comb ¼
Wg þ WST DHf
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 as Wg ¼ Cp ½2ðh3 h4 Þ ðh2 h1 Þ
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 1 Wg ¼ Cp ðn þ 1ÞRGT Tmax 1 T1 AC 1 =AC ð16Þ RGT Dividing by CpT0, the dimensionless specific power-output becomes wg ¼ ðn þ 1ÞRGT
RGT
where !AC=AC 1 and
ð
RGT
AC
AC Þ
¼ 1 1
RGT
ð17Þ
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The fuel input or heat input (Hf or Qin) per unit mass of the cycle for the single stage with full reheat is given by
Qin ¼ DHf;CC þ DHf;RGT ð18Þ For a perfect gas, it may be expressed as T2 DHf ¼ Qin ¼ CP Tmax T1 AC þ RGT Tmax AC For ‘n’ reheat stage, it becomes AC Qin ¼ CP Tmax T1 T2 þ nRGT Tmax AC
RGT
ð19Þ
RGT
ð20Þ
Dividing by CpT0, 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
Wg ðn þ 1ÞRGT RGT AC =AC ¼ qin 1 AC =AC þ nRGT RGT
ð22Þ
Using Eqs. (12), (13), (17) and (21), the first law efficiency of the combined cycle may be expressed as
ðn þ 1ÞRGT RGT AC =2;ST w4 1 ; comb ¼ ð23Þ ½ 1 AC =AC þ nRGT RGT 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
Wg þ WST 1;Comb ¼ ef Carnot
Using Eqs. (23) and (9) in Eq. (24),
ðn þ 1ÞRGT RGT AC =AC þ 2;ST w4 2 ; comb ¼ ½ 1 AC =AC þ nRGT RGT ð 1Þ
ð24Þ
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 compression ratio 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 where
out
=1 pin p pin
185
ð26Þ ¼1
Dp p
The quantity p/p is known as the relative pressure-drop and b may be called the pressure-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 pRi R
ð28Þ
pgo ¼ g pgi
ð29Þ
Combining Eqs. (27)–(29), we have p3 pRo p2 ¼ CC R g ¼ CC R g AC pR i pg i pg o
ð30Þ
Now p3 pR ¼ o ¼ AC pR i pg i For a system with one stage of reheat, ðRGT Þ2 ¼ CC R g AC
ð31Þ
1=2 RGT ¼ CC R g AC
ð32Þ
For two reheat-stages, 1=3 RGT ¼ CC 2R g AC
ð33Þ
For n reheat-stages, 1=ðnþ1Þ RGT ¼ CC nR g AC
ð34Þ
The traditional first-law efficiency of a steam turbine cycle is 1;ST ¼
WST QST
ð35Þ
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Its second-law efficiency has been defined by Eq. (11). Thus the ratio of its secondlaw 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 T4 and leaves at Tex, then 2;ST WST QST QST ¼ ¼ 1;ST e4 WST e4 For a constant-pressure process, by dividing by CpT0 2;ST 4 ex ¼ 1;ST 4 1 ‘n 4
ð36Þ
ð37Þ
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 bottoming 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 e2 e1 2;AC ¼ ð38Þ WAC For frictionless reversible adiabatic or isothermal compressions, no entropy is generated or exergy destroyed and 2,AC=1. In a real compressor of adiabatic efficiency AC, for an infinitesimal adiabatic increase in pressure dp, the temperature increase dT is greater than the isentropic value dTs. dT ¼
dTs AC
ð39Þ
For a perfect gas using the isentropic relation, we have dTs -1 dp ¼ T p
ð40Þ
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187
The canonical relation is ds ¼ Cp
dTs dp R T p
ð41Þ
Using Eqs. (39) and (41), the entropy generated during the compression process is dsgen ¼ Cp
dTs dp R AC p
Using Eqs. (40) and (42), we have 1 AC dp dsgen ¼ R p AC
ð42Þ
ð43Þ
The exergy destroyed is obtained after multiplying Eq. (43) by T0 and then integrating. After non-dimensionalizing by dividing with CpT0, the dimensionless exergy destruction may be given as 1 AC wD;AC ¼ ‘nr ð44Þ AC Eq. (44) accounts for the exergy destroyed within the compressor. The compressor work for the infinitesimal adiabatic stage is CpdT. After using Eqs. (39) and (40) for a perfect gas, the compressor work in dimensionless form may be given as ð 1 2 T dp ð45Þ wAC ¼ AC 1 T0 p Unlike the exergy destroyed, this depends on the local temperature. The compression 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 1 AC ‘nr 2;ACad ¼ 1 ð47Þ r1=AC 1 AC 5.2. Combustion chamber (CC) The heat addition in the combustion chamber (Hf,CC) may be defined as
DHf;CC ¼
Q
m
¼ Cp ðT3 T2 Þ
After dividing Eq. (48) by CpT0, it may be expressed as
ð48Þ
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Dhf;CC ¼ 2
ð49Þ
The exergy associated with Hf,CC is ef;CC ¼ DHf;CC ð1 1=Þ
ð50Þ
Using Eq. (49) and dividing Eq. (50) by CpT0, the dimensionless exergy associated with fuel may be obtained as wf;CC ¼
ð 1Þð 2 Þ
ð51Þ
The increase in exergy per unit mass of fuel is given by e 3 e 2 ¼ ð h3 h2 Þ T 0 ð s 3 s 2 Þ
ð52Þ
After dividing by CpT0 and using Eqs. (41), (49) and (26), it may further be expressed as ð53Þ w3 w2 ¼ 2 ‘n þ ‘n CC 2 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 ‘n CC 1 2
ð54Þ
The second-law efficiency for the combustion chamber is the ratio of the increased exergy over the exergy input and is given by w3 w2 2;CC ¼ ð55Þ wf;CC Using Eqs. (49) and (53), 2 ‘n þ ‘n CC 2 2;CC ¼ ð 2 Þð 1Þ
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 isentropic value dTs. RGT ¼
dT dTs
ð57Þ
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189
Using Eqs. (40), (41) and (43), we see that the entropy generated in the adiabatic stage is dsgen ¼ ð1 RGT ÞR
dp p
ð58Þ
By multiplying Eq. (58) by T0 and integrating, and, thereafter dividing by CpT0, the exergy destruction may be obtained as wD;RGT ¼ ð1 RGT Þ‘nr
ð59Þ
It accounts only for exergy destroyed within the turbine but not for reheat pressure-losses or heat-transfer losses. The expansion work CpdT, after using Eqs. (57) and (40), may be expressed as ð ð 1Þ 4 T dp ð60Þ wRGT ¼ RGT 3 T0 p and depends on the pressure–temperature path. For the adiabatic expansion starting at T3 after integrating Eq. (60) and using (=T3/T0), it may also be expressed as wRGT;ad ¼ ð1 rRGT 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 wRGT 2;RGT ¼ ð62Þ w3 w2 þ wRGT Using Eqs. (61), (53) and (62), the second-law efficiency of the expansion process in the gas turbine cycle may be given as ð1 RGT Þr‘nr 1 2;RGTad ¼ 1 þ ð63Þ ð1 rRGT Þ 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 be recovered.
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 ¼0 @AC
ð64Þ
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This gives
1 ðAC Þopt ¼ AC RGT
2ð1 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 compressor 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 combined 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 efficiency 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 T4. 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 difference, the net gas-turbine work output drops. The absolute magnitude of exergy destroyed 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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A. Khaliq, S.C. Kaushik / Applied Energy 78 (2004) 179–197 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 2 4 8 16 32 64 128
1.000 1.219 1.485 1.811 2.208 2.691 3.281 3.999
0.900 0.890 0.880 0.870 0.860 0.850 0.832 0.820
0.000 0.022 0.043 0.066 0.088 0.110 0.132 0.154
0.900 0.910 0.920 0.929 0.937 0.945 0.951 0.957
0.950 1.151 1.369 1.628 1.76 2.303 2.739 3.257
0.985 1.041 1.094 1.149 1.175 1.269 1.333 1.401
0.850 0.862 0.872 0.883 0.894 0.904 0.912 0.920
0.000 0.029 0.059 0.089 0.118 0.148 0.178 0.207
0.995 0.955 0.941 0.924 0.903 0.876 0.844 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) 0 1 2 3 4 5 6 7
wD,RGT
R=1.0
R=0.98
R=0.96
0.1485 0.2257 0.2203 0.2107 0.2023 0.1954 0.1908 0.1858
0.1485 0.2331 0.2277 0.2223 0.2201 0.2177 0.2166 0.2173
0.1485 0.2380 0.2389 0.2407 0.2441 0.2497 0.2565 0.2643
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 is known from the second-law analysis. The exergy destructions due to heat-transfer irreversibility (HRSG), condenserheat rejection, irreversibilities of the steam turbine and pump, and the first-law efficiency 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 cycle efficiency 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
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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 16 32 64 128
1.920 2.42 3.048 3.840 4.838
3.413 3.227 3.050 2.880 2.727
51.1 52.3 50.7 46.7 38.0
3.840 3.630 3.433 3.245 3.068
53.70 55.76 56.36 54.80 49.00
4.267 4.034 3.814 3.606 3.409
55.80 58.20 59.20 58.80 56.50
4.69 4.438 4.195 3.960 3.750
57.5 60.0 61.45 61.65 60.30
5.0 4.84 4.57 4.12 3.72
59.0 61.57 63.23 63.80 63.20
(b) 8 16 32 64 128
1.920 2.42 3.048 3.840 4.838
3.413 3.227 3.050 2.880 2.727
68.13 69.73 67.6 62.26 50.66
3.840 3.630 3.433 3.245 3.068
69.11 71.76 72.53 70.52 63.06
4.267 4.034 3.814 3.606 3.409
69.75 72.75 74.0 73.5 70.62
4.69 4.438 4.195 3.960 3.750
70.30 73.35 75.12 75.36 73.71
5.0 4.84 4.57 4.12 3.72
70.80 73.80 75.80 76.56 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 16 32 64 128
27.8 33.0 35.95 36.7 34.4
34.75 41.25 44.93 45.87 43.00
28.00 25.17 23.25 22.40 22.00
43.82 40.18 37.82 37.1,6 37.00
55.85 58.17 59.2 58.8 56.40
69.81 72.71 74.00 73.50 70.50
80.00 80.00 80.00 80.00 80.00
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 the specific 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 ratio increases the overall thermal efficiency at a given maximum temperature. However increasing the pressure ratio beyond a certain value at any given maximum
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Table 5 Effect of pressure ratio (AC) and cycle temperature ratio () on exergy destruction and second law efficiency of combustion chamber (CC) for two reheats AC
1 2 4 8 16 32 64 128
=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
1.627 1.484 1.353 1.236 1.137 1.060 1.010 0.990
0.133 0.163 0.192 0.214 0.215 0.176 0.058 0.028
1.717 1.568 1.430 1.303 1.190 1.103 1.036 0.997
0.126 1.550 0.186 0.210 0.222 0.207 0.134 0.100
1.800 1.647 1.502 1.368 1.248 1.148 1.068 1.014
0.119 0.146 0.176 0.204 0.223 0.222 0.180 0.061
1.877 1.719 1.570 1.439 1.305 1.195 1.103 1.036
0.112 0.138 0.167 0.196 0.219 0.228 0.206 0.126
1.949 1.423 1.634 1.490 1.359 1.241 1.141 1.062
0.1066 0.1317 0.1590 0.1880 0.2140 0.2300 0.2210 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 4.5 5.0 5.5 6.0 6.5
30.30 33.70 36.00 37.60 38.85 39.80
40.40 43.33 45.00 46.95 46.62 47.05
20.50 22.57 23.30 23.90 24.39 24.86
35.60 37.90 37.95 37.86 37.63 37.50
50.80 56.27 59.30 61.50 63.24 64.66
67.70 72.40 74.12 75.18 75.89 76.40
75.00 77.70 80.00 81.80 83.33 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
Wg
wg+w4
wComb
qin
Carnotqin
0 1 2 3 4 5
43.50 37.28 36.7 36.2 35.9 35.7
9.00 66.80 69.57 70.63 71.40 71.90
52.50 57.90 59.77 60.33 60.75 61.00
0.950 1.403 1.644 1.76 1.828 1.865
1.500 2.513 3.109 3.425 3.63 3.756
1.20 2.18 2.67 2.926 3.089 3.186
2.100 3.762 4.469 4.85 5.086 5.224
1.700 3.00 3.575 3.880 4.068 4.180
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 much lower pressure-ratio than the point of maximum efficiency for the same maximum temperature.
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Table 8 Exergy destruction as a percentage of heat added, in the components of the steam-turbine cycle: T0=291 K, Tex=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 2.25 2.50 2.75 3.00 3.25
73 81 85 88 90 91
13 18 16 17 16 13
6 5 6 5 5 4
4 6 5 6 7 8
49 52 58 61 63 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 2;ST 1;ST
ex=1.5 2;ST 1;ST
ex=2.0 2;ST 1;ST
ex=2.5 2;ST 1;ST
ex=3.0 2;ST 1;ST
ex=3.5 2;ST 1;ST
ex=40 2;ST 1;ST
2 3 4 5 6 7 8 9 10 11
3.258 2.218 1.859 1.673 1.558 1.479 – – – –
1.629 1.664 1.459 1.464 1.4026 1.356 – – – –
1.109 1.109 1.239 1.255 1.246 1.233 – – – –
0.5540 0.5540 0.9290 1.0457 1.0909 1.1099 – – – –
– – 0.6196 0.8366 0.9350 0.9866 1.0160 1.0339 1.0450 1.0523
– – 0.3098 0.6274 0.7792 0.8633 0.9145 0.9478 0.9705 0.9865
– – – 0.4180 0.6230 0.7400 0.8129 0.8616 0.8958 0.9207
Thus, a cursory inspection of the efficiency indicates that the gas-turbine cycle efficiency can be improved by increasing the pressure ratio, or increasing the turbine’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 performance. The exergy balance or second-law approach presented facilitates the design and optimization of complex cycles by pinpointing and quantifying the losses. By
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195
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.
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 by increasing the specific power-output reduces the sensitivity of the cycle to component losses.
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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 T4 for a fixed exhaust-gas temperature Tex. To find this correlation, calculations were done for several values of the temperature T4. In each case, the steam-turbine cycle pressure and peak temperature T5,ST were first determined by setting the pinch point (saturation) and maximum steamtemperatures 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 T4 Tsat þ 50 h5;st hsat;liq ¼ T4 Tex 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.
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1. 2. 3. 4. 5.
197
Ambient temperature T0=291 K Exhaust temperature Tex=420 K Condenser pressure 0.045 bar (304 K) Steam turbine and feed water pump have efficiencies 90 and 70% respectively. Saturation temperature (Tsat)=Tex22 C.
For each T4, 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 [1] Sieppel C, Bereuter R. The theory of combined steam and gas-turbine installation. Brown Boveri Review 1960;47:83–799. [2] Czermak H, Wunsch A. The 125 MW combined cycle plant design features, plan performance and operating experience paper no. 82 GT-323]. ASME; 1982. [3] Wunsch A. Highest efficiencies possible by converting gas-turbine plants into combined cycle plants. Brown Boveri Review 1985;10:455–63. [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 insight on inter-cooling and reheat gas-turbine cycles. Proc Instn Mech Engnrs J Power Energy 2001; 215A. [8] Andriani R, Ghezzi U, Anntoni LFG. Jet engines with heat addition during expansion: a performance analysis paper 99-0744]. AIAA; 1999. [9] Polyzakis A. Industrial gas-turbine for combined cycle plant. MSc thesis, Cranfield University, 1995. [10] Bejan A. Fundamentals of exergy analysis, entropy generation minimization, and the generation’ of flow architecture. Int J Energy Res 2002;26:545–65.
Second-law based thermodynamic analysis of Brayton/Rankine combined power cycle with reheat A. Khaliqa,*, S.C. Kaushikb a
Department of Mechanical Engineering, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi 110025, India b Centre 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 secondlaw 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 temperatureratio, number of reheats and cycle pressure-drop on the combined cycle performance parameters 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 combined 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 * Corresponding author. E-mail addresses: [email protected]. 0306-2619/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2003.08.002
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Nomenclature Cp e h n p Q R Sge s W w AC 1,g 1,Comb 2,Comb hf Hf
Specific heat at constant pressure (kJ/kg K) Specific exergy (kJ/kg) Specific enthalpy (kJ/kg) Number of reheat stages Pressure (kPa) Heat per unit mass of fuel (KJ/kg) Gas constant (KJ/kg-K) Entropy generation rate (W/K) Specific entropy (KJ/kg) Work per unit mass of gas (KJ/kg) Dimensionless specific exergy/work (w=e/CpT0) Pressure ratio across the compressor Ratio of specific heats First-law efficiency of gas-turbine cycle First-law efficiency of combined cycle Second-law efficiency of combined cycle Maximum to minimum cycle temperature ratio (=T3/T0) Dimensionless heat-input (Hf/CpT0) Heat input or enthalpy of reaction at standard condition (KJ/kg)
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 thermodynamic analysis for a practicable Brown Boveri 125 MW combined gas–steam turbine power plant. Wunsch [3] claimed that the efficiencies of combined gas–steam plants were more influenced by the gas-turbine parameters like maximum temperature 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 efficiency. 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 combined 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 good performance if suitably designed, but if the highest gas-turbine temperatures are used, expensive fuel must be utilized.
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181
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 considered 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 temperature (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 combined-cycle characteristics. Andriani et al. [8] carried out the analysis of a gas turbine 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 applications. 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 complication 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 thermodynamic evaluation of energy conservation in thermal power cycles. It became apparent to the current authors that, although there was sufficient literature 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-temperature loop, whereas the steam plant forms the low-temperature loop. The connecting link 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]
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Fig. 1. Schematic diagram of the combined Brayton/Rankine power cycle with reheat.
EW ¼
n X X X EQ þ me me ED i
i¼1
ð1Þ
out
in
for a single-stream flow
EW ¼ EQ þ mein meout meD
ð2Þ
After making exergy balances using Eq. (2) for the compressor, reheat turbine and combustion chamber, the following expressions can be obtained WAC ¼ ðe2 e1 Þ þ eD;AC
ð3Þ
ef;CC ¼ ðe3 e2 Þ þ eD;CC
ð4Þ
eRGT ¼ ðe4 e3 Þ þ WRGT þ eD;RGT
ð5Þ
where WAC and WRGT are the work done per unit mass for the compressor and reheat gas-turbine respectively and ‘e’ is the specific exergy The net work-output of the gas-turbine cycle is
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Wg ¼ WRGT WAC
ð6Þ
The first law efficiency of the gas-turbine cycle is given by 1;g ¼
Wg DHf
ð7Þ
The total exergy of the fuel input for the gas turbine cycle with reheat is ef ¼ ef;CC þ eRGT
If we define maximum to minimum cycle temperature ratio as then the exergy associated with the fuel can be expressed as 1 ef ¼ Carnot DHf ¼ 1 DHf
ð8Þ Tmax T3 ¼ , ¼ Tmin T0 ð9Þ
The second-law efficiency of the gas-turbine cycle may be defined as 2;g ¼
Wg 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. WST ¼ 2;ST e4
ð11Þ
Dividing by CpT0, Eq. (11) becomes WST ¼ 2;ST W4
ð12Þ
The first-law efficiency of the combined power cycle is given by 1;Comb ¼
Wg þ WST DHf
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 as Wg ¼ Cp ½2ðh3 h4 Þ ðh2 h1 Þ
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 1 Wg ¼ Cp ðn þ 1ÞRGT Tmax 1 T1 AC 1 =AC ð16Þ RGT Dividing by CpT0, the dimensionless specific power-output becomes wg ¼ ðn þ 1ÞRGT
RGT
where !AC=AC 1 and
ð
RGT
AC
AC Þ
¼ 1 1
RGT
ð17Þ
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The fuel input or heat input (Hf or Qin) per unit mass of the cycle for the single stage with full reheat is given by
Qin ¼ DHf;CC þ DHf;RGT ð18Þ For a perfect gas, it may be expressed as T2 DHf ¼ Qin ¼ CP Tmax T1 AC þ RGT Tmax AC For ‘n’ reheat stage, it becomes AC Qin ¼ CP Tmax T1 T2 þ nRGT Tmax AC
RGT
ð19Þ
RGT
ð20Þ
Dividing by CpT0, 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
Wg ðn þ 1ÞRGT RGT AC =AC ¼ qin 1 AC =AC þ nRGT RGT
ð22Þ
Using Eqs. (12), (13), (17) and (21), the first law efficiency of the combined cycle may be expressed as
ðn þ 1ÞRGT RGT AC =2;ST w4 1 ; comb ¼ ð23Þ ½ 1 AC =AC þ nRGT RGT 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
Wg þ WST 1;Comb ¼ ef Carnot
Using Eqs. (23) and (9) in Eq. (24),
ðn þ 1ÞRGT RGT AC =AC þ 2;ST w4 2 ; comb ¼ ½ 1 AC =AC þ nRGT RGT ð 1Þ
ð24Þ
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 compression ratio 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 where
out
=1 pin p pin
185
ð26Þ ¼1
Dp p
The quantity p/p is known as the relative pressure-drop and b may be called the pressure-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 pRi R
ð28Þ
pgo ¼ g pgi
ð29Þ
Combining Eqs. (27)–(29), we have p3 pRo p2 ¼ CC R g ¼ CC R g AC pR i pg i pg o
ð30Þ
Now p3 pR ¼ o ¼ AC pR i pg i For a system with one stage of reheat, ðRGT Þ2 ¼ CC R g AC
ð31Þ
1=2 RGT ¼ CC R g AC
ð32Þ
For two reheat-stages, 1=3 RGT ¼ CC 2R g AC
ð33Þ
For n reheat-stages, 1=ðnþ1Þ RGT ¼ CC nR g AC
ð34Þ
The traditional first-law efficiency of a steam turbine cycle is 1;ST ¼
WST QST
ð35Þ
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Its second-law efficiency has been defined by Eq. (11). Thus the ratio of its secondlaw 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 T4 and leaves at Tex, then 2;ST WST QST QST ¼ ¼ 1;ST e4 WST e4 For a constant-pressure process, by dividing by CpT0 2;ST 4 ex ¼ 1;ST 4 1 ‘n 4
ð36Þ
ð37Þ
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 bottoming 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 e2 e1 2;AC ¼ ð38Þ WAC For frictionless reversible adiabatic or isothermal compressions, no entropy is generated or exergy destroyed and 2,AC=1. In a real compressor of adiabatic efficiency AC, for an infinitesimal adiabatic increase in pressure dp, the temperature increase dT is greater than the isentropic value dTs. dT ¼
dTs AC
ð39Þ
For a perfect gas using the isentropic relation, we have dTs -1 dp ¼ T p
ð40Þ
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187
The canonical relation is ds ¼ Cp
dTs dp R T p
ð41Þ
Using Eqs. (39) and (41), the entropy generated during the compression process is dsgen ¼ Cp
dTs dp R AC p
Using Eqs. (40) and (42), we have 1 AC dp dsgen ¼ R p AC
ð42Þ
ð43Þ
The exergy destroyed is obtained after multiplying Eq. (43) by T0 and then integrating. After non-dimensionalizing by dividing with CpT0, the dimensionless exergy destruction may be given as 1 AC wD;AC ¼ ‘nr ð44Þ AC Eq. (44) accounts for the exergy destroyed within the compressor. The compressor work for the infinitesimal adiabatic stage is CpdT. After using Eqs. (39) and (40) for a perfect gas, the compressor work in dimensionless form may be given as ð 1 2 T dp ð45Þ wAC ¼ AC 1 T0 p Unlike the exergy destroyed, this depends on the local temperature. The compression 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 1 AC ‘nr 2;ACad ¼ 1 ð47Þ r1=AC 1 AC 5.2. Combustion chamber (CC) The heat addition in the combustion chamber (Hf,CC) may be defined as
DHf;CC ¼
Q
m
¼ Cp ðT3 T2 Þ
After dividing Eq. (48) by CpT0, it may be expressed as
ð48Þ
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Dhf;CC ¼ 2
ð49Þ
The exergy associated with Hf,CC is ef;CC ¼ DHf;CC ð1 1=Þ
ð50Þ
Using Eq. (49) and dividing Eq. (50) by CpT0, the dimensionless exergy associated with fuel may be obtained as wf;CC ¼
ð 1Þð 2 Þ
ð51Þ
The increase in exergy per unit mass of fuel is given by e 3 e 2 ¼ ð h3 h2 Þ T 0 ð s 3 s 2 Þ
ð52Þ
After dividing by CpT0 and using Eqs. (41), (49) and (26), it may further be expressed as ð53Þ w3 w2 ¼ 2 ‘n þ ‘n CC 2 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 ‘n CC 1 2
ð54Þ
The second-law efficiency for the combustion chamber is the ratio of the increased exergy over the exergy input and is given by w3 w2 2;CC ¼ ð55Þ wf;CC Using Eqs. (49) and (53), 2 ‘n þ ‘n CC 2 2;CC ¼ ð 2 Þð 1Þ
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 isentropic value dTs. RGT ¼
dT dTs
ð57Þ
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189
Using Eqs. (40), (41) and (43), we see that the entropy generated in the adiabatic stage is dsgen ¼ ð1 RGT ÞR
dp p
ð58Þ
By multiplying Eq. (58) by T0 and integrating, and, thereafter dividing by CpT0, the exergy destruction may be obtained as wD;RGT ¼ ð1 RGT Þ‘nr
ð59Þ
It accounts only for exergy destroyed within the turbine but not for reheat pressure-losses or heat-transfer losses. The expansion work CpdT, after using Eqs. (57) and (40), may be expressed as ð ð 1Þ 4 T dp ð60Þ wRGT ¼ RGT 3 T0 p and depends on the pressure–temperature path. For the adiabatic expansion starting at T3 after integrating Eq. (60) and using (=T3/T0), it may also be expressed as wRGT;ad ¼ ð1 rRGT 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 wRGT 2;RGT ¼ ð62Þ w3 w2 þ wRGT Using Eqs. (61), (53) and (62), the second-law efficiency of the expansion process in the gas turbine cycle may be given as ð1 RGT Þr‘nr 1 2;RGTad ¼ 1 þ ð63Þ ð1 rRGT Þ 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 be recovered.
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 ¼0 @AC
ð64Þ
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This gives
1 ðAC Þopt ¼ AC RGT
2ð1 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 compressor 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 combined 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 efficiency 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 T4. 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 difference, the net gas-turbine work output drops. The absolute magnitude of exergy destroyed 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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A. Khaliq, S.C. Kaushik / Applied Energy 78 (2004) 179–197 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 2 4 8 16 32 64 128
1.000 1.219 1.485 1.811 2.208 2.691 3.281 3.999
0.900 0.890 0.880 0.870 0.860 0.850 0.832 0.820
0.000 0.022 0.043 0.066 0.088 0.110 0.132 0.154
0.900 0.910 0.920 0.929 0.937 0.945 0.951 0.957
0.950 1.151 1.369 1.628 1.76 2.303 2.739 3.257
0.985 1.041 1.094 1.149 1.175 1.269 1.333 1.401
0.850 0.862 0.872 0.883 0.894 0.904 0.912 0.920
0.000 0.029 0.059 0.089 0.118 0.148 0.178 0.207
0.995 0.955 0.941 0.924 0.903 0.876 0.844 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) 0 1 2 3 4 5 6 7
wD,RGT
R=1.0
R=0.98
R=0.96
0.1485 0.2257 0.2203 0.2107 0.2023 0.1954 0.1908 0.1858
0.1485 0.2331 0.2277 0.2223 0.2201 0.2177 0.2166 0.2173
0.1485 0.2380 0.2389 0.2407 0.2441 0.2497 0.2565 0.2643
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 is known from the second-law analysis. The exergy destructions due to heat-transfer irreversibility (HRSG), condenserheat rejection, irreversibilities of the steam turbine and pump, and the first-law efficiency 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 cycle efficiency 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
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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 16 32 64 128
1.920 2.42 3.048 3.840 4.838
3.413 3.227 3.050 2.880 2.727
51.1 52.3 50.7 46.7 38.0
3.840 3.630 3.433 3.245 3.068
53.70 55.76 56.36 54.80 49.00
4.267 4.034 3.814 3.606 3.409
55.80 58.20 59.20 58.80 56.50
4.69 4.438 4.195 3.960 3.750
57.5 60.0 61.45 61.65 60.30
5.0 4.84 4.57 4.12 3.72
59.0 61.57 63.23 63.80 63.20
(b) 8 16 32 64 128
1.920 2.42 3.048 3.840 4.838
3.413 3.227 3.050 2.880 2.727
68.13 69.73 67.6 62.26 50.66
3.840 3.630 3.433 3.245 3.068
69.11 71.76 72.53 70.52 63.06
4.267 4.034 3.814 3.606 3.409
69.75 72.75 74.0 73.5 70.62
4.69 4.438 4.195 3.960 3.750
70.30 73.35 75.12 75.36 73.71
5.0 4.84 4.57 4.12 3.72
70.80 73.80 75.80 76.56 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 16 32 64 128
27.8 33.0 35.95 36.7 34.4
34.75 41.25 44.93 45.87 43.00
28.00 25.17 23.25 22.40 22.00
43.82 40.18 37.82 37.1,6 37.00
55.85 58.17 59.2 58.8 56.40
69.81 72.71 74.00 73.50 70.50
80.00 80.00 80.00 80.00 80.00
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 the specific 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 ratio increases the overall thermal efficiency at a given maximum temperature. However increasing the pressure ratio beyond a certain value at any given maximum
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Table 5 Effect of pressure ratio (AC) and cycle temperature ratio () on exergy destruction and second law efficiency of combustion chamber (CC) for two reheats AC
1 2 4 8 16 32 64 128
=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
1.627 1.484 1.353 1.236 1.137 1.060 1.010 0.990
0.133 0.163 0.192 0.214 0.215 0.176 0.058 0.028
1.717 1.568 1.430 1.303 1.190 1.103 1.036 0.997
0.126 1.550 0.186 0.210 0.222 0.207 0.134 0.100
1.800 1.647 1.502 1.368 1.248 1.148 1.068 1.014
0.119 0.146 0.176 0.204 0.223 0.222 0.180 0.061
1.877 1.719 1.570 1.439 1.305 1.195 1.103 1.036
0.112 0.138 0.167 0.196 0.219 0.228 0.206 0.126
1.949 1.423 1.634 1.490 1.359 1.241 1.141 1.062
0.1066 0.1317 0.1590 0.1880 0.2140 0.2300 0.2210 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 4.5 5.0 5.5 6.0 6.5
30.30 33.70 36.00 37.60 38.85 39.80
40.40 43.33 45.00 46.95 46.62 47.05
20.50 22.57 23.30 23.90 24.39 24.86
35.60 37.90 37.95 37.86 37.63 37.50
50.80 56.27 59.30 61.50 63.24 64.66
67.70 72.40 74.12 75.18 75.89 76.40
75.00 77.70 80.00 81.80 83.33 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
Wg
wg+w4
wComb
qin
Carnotqin
0 1 2 3 4 5
43.50 37.28 36.7 36.2 35.9 35.7
9.00 66.80 69.57 70.63 71.40 71.90
52.50 57.90 59.77 60.33 60.75 61.00
0.950 1.403 1.644 1.76 1.828 1.865
1.500 2.513 3.109 3.425 3.63 3.756
1.20 2.18 2.67 2.926 3.089 3.186
2.100 3.762 4.469 4.85 5.086 5.224
1.700 3.00 3.575 3.880 4.068 4.180
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 much lower pressure-ratio than the point of maximum efficiency for the same maximum temperature.
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Table 8 Exergy destruction as a percentage of heat added, in the components of the steam-turbine cycle: T0=291 K, Tex=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 2.25 2.50 2.75 3.00 3.25
73 81 85 88 90 91
13 18 16 17 16 13
6 5 6 5 5 4
4 6 5 6 7 8
49 52 58 61 63 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 2;ST 1;ST
ex=1.5 2;ST 1;ST
ex=2.0 2;ST 1;ST
ex=2.5 2;ST 1;ST
ex=3.0 2;ST 1;ST
ex=3.5 2;ST 1;ST
ex=40 2;ST 1;ST
2 3 4 5 6 7 8 9 10 11
3.258 2.218 1.859 1.673 1.558 1.479 – – – –
1.629 1.664 1.459 1.464 1.4026 1.356 – – – –
1.109 1.109 1.239 1.255 1.246 1.233 – – – –
0.5540 0.5540 0.9290 1.0457 1.0909 1.1099 – – – –
– – 0.6196 0.8366 0.9350 0.9866 1.0160 1.0339 1.0450 1.0523
– – 0.3098 0.6274 0.7792 0.8633 0.9145 0.9478 0.9705 0.9865
– – – 0.4180 0.6230 0.7400 0.8129 0.8616 0.8958 0.9207
Thus, a cursory inspection of the efficiency indicates that the gas-turbine cycle efficiency can be improved by increasing the pressure ratio, or increasing the turbine’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 performance. The exergy balance or second-law approach presented facilitates the design and optimization of complex cycles by pinpointing and quantifying the losses. By
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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.
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 by increasing the specific power-output reduces the sensitivity of the cycle to component losses.
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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 T4 for a fixed exhaust-gas temperature Tex. To find this correlation, calculations were done for several values of the temperature T4. In each case, the steam-turbine cycle pressure and peak temperature T5,ST were first determined by setting the pinch point (saturation) and maximum steamtemperatures 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 T4 Tsat þ 50 h5;st hsat;liq ¼ T4 Tex 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.
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1. 2. 3. 4. 5.
197
Ambient temperature T0=291 K Exhaust temperature Tex=420 K Condenser pressure 0.045 bar (304 K) Steam turbine and feed water pump have efficiencies 90 and 70% respectively. Saturation temperature (Tsat)=Tex22 C.
For each T4, 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.
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