Power and efficiency optimization for combined Brayton and inverse Brayton cycles

Applied Thermal Engineering 29 (2009) 2885–2894

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Power and efficiency optimization for combined Brayton and inverse Brayton cycles Wanli Zhang, Lingen Chen *, Fengrui Sun Postgraduate School, Naval University of Engineering, Wuhan 430033, PR China

a r t i c l e

i n f o

Article history: Received 21 February 2007 Accepted 16 February 2009 Available online 21 February 2009 Keywords: Brayton cycle Inverse Brayton cycle Combined cycle Power output Efficiency Area allocation Pressure drop loss Finite time thermodynamics Thermodynamic optimization

a b s t r a c t A thermodynamic model for open combined Brayton and inverse Brayton cycles is established considering the pressure drops of the working fluid along the flow processes and the size constraints of the real power plant using finite time thermodynamics in this paper. There are 11 flow resistances encountered by the gas stream for the combined Brayton and inverse Brayton cycles. Four of these, the friction through the blades and vanes of the compressors and the turbines, are related to the isentropic efficiencies. The remaining flow resistances are always present because of the changes in flow cross-section at the compressor inlet of the top cycle, combustion inlet and outlet, turbine outlet of the top cycle, turbine outlet of the bottom cycle, heat exchanger inlet, and compressor inlet of the bottom cycle. These resistances control the air flow rate and the net power output. The relative pressure drops associated with the flow through various cross-sectional areas are derived as functions of the compressor inlet relative pressure drop of the top cycle. The analytical formulae about the relations between power output, thermal conversion efficiency, and the compressor pressure ratio of the top cycle are derived with the 11 pressure drop losses in the intake, compression, combustion, expansion, and flow process in the piping, the heat transfer loss to the ambient, the irreversible compression and expansion losses in the compressors and the turbines, and the irreversible combustion loss in the combustion chamber. The performance of the model cycle is optimized by adjusting the compressor inlet pressure of the bottom cycle, the air mass flow rate and the distribution of pressure losses along the flow path. It is shown that the power output has a maximum with respect to the compressor inlet pressure of the bottom cycle, the air mass flow rate or any of the overall pressure drops, and the maximized power output has an additional maximum with respect to the compressor pressure ratio of the top cycle. When the optimization is performed with the constraints of a fixed fuel flow rate and the power plant size, the power output and efficiency can be maximized again by properly allocating the fixed overall flow area among the compressor inlet of the top cycle and the turbine outlet of the bottom cycle. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Due to the increased request to the effective of thermodynamic cycles, combined cycle model attracts more and more people’s attention in recent years. It is estimated that, owing to the advances in the technologies of the individual components, the thermal efficiency of large combined cycle plant consisting of a gas turbine and a stream bottoming cycle exceeded 55% (based on the lower heating value of the fuel) several years ago and is now at approximately 60% [1]. More and more new configurations of combined cycle power plant have been provided recently. Frost et al. [2] proposed a hybrid gas turbine cycle (Braysson cycle) based on the conventional Brayton cycle for high-temperature heat addition process while adoption the Ericsson cycle for the low temperature heat rejection process, and performed first law analysis based on energy balance. Zheng et al. [3] performed second law * Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen). 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.02.011

analysis of the Braysson cycle based on exergy balance. Furthermore, the performance optimization of an endoreversible Braysson cycle with heat resistance losses in the hot-side and cold-side heat exchangers were performed by using finite time thermodynamics [4]. Fujii et al. [5] studied a bottom cycle constructed from an expander followed by an intercooled compression processes (mirror arrangement) in 2001. They limited the bottom cycle expansion pressure to 0.25 bar to avoid a rapid increase in gas flow axial velocity. They also proposed the use of two parallel inverse Brayton cycles instead of one. Bianchi et al. [6] examined an inverted Brayton cycle in which the processes were expansion, cooling at constant pressure, and compression to atmospheric pressure followed by a heat recovery heat exchanger in 2002. Agnew et al. [7] proposed a simple inverse Brayton cycle arrangement that consisted of an expansion and cooling at constant pressure followed by a recompression process. It revealed the range of bottom cycle operated and stressed a value of 0.5 bar for optimum performance. This value of pressure also contributed to reducing manufacturing difficulties and extra cost. Zhang et al. [8] studied combined

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Nomenclature A a C h K L _ m P Q Q_ Q R s T v w _ W  W x

area (m2) ratio of the outermost equivalent flow cross-sections specific heat [kJ/(kg K)]/constant enthalpy (kJ/kg) contraction pressure loss coefficient excess air ratio mass flow rate (kg/s) pressure (MPa) heat transfer quantity (kJ/kg) heat transfer rate (kW) dimensionless heat transfer rate ideal gas constant (kJ/(kg K)) entropy (kJ/(kg K)) temperature (K) mean velocity (m/s) specific work (kJ/kg) power (kW) dimensionless power flow area allocation ratio

Greek symbols b pressure ratio D variation e effectiveness of the heat exchanger c ratio of specific heats g efficiency

Brayton and inverse Brayton cycles using exergy analysis. Alabdoadaim et al. [9–11] further presented new configurations of combined Brayton and inverse Brayton cycles and performed first law analysis based on energy balance. Finite time thermodynamics (FTT) and entropy generation minimization were powerful tools for the performance analysis and optimization of finite time processes and finite size devices [12– 23]. Much work has concerned the FTT performance analysis and optimization for closed, simple, regenerated, intercooled, and intercooled and regenerated, endoreversible and irreversible Brayton cycles by taking the power, specific power, power density, efficiency, and ecological optimization as objectives with considerations of the heat transfer irreversibility and/or internal irreversibilities. For the closed Brayton cycles, the principle of optimally dividing a finite heat exchanger inventory between the hot and cold end of the power plant was used [24,25]. In practice industry, open-cycle gas turbine power plants are widely applied. Radcenco et al. [26,27] optimized the performance of an open-cycle simple gas turbine power plant by incorporating the irreversibilities due to the various pressure drops distributed along the flow path into the power plant model. Chen et al. [28] optimized the power and efficiency of an open-cycle regenerated gas turbine power plant by using the similar method. Wang et al. [29] optimized the power and efficiency of an open-cycle intercooled gas turbine power plant by using the similar method. The analogy between the irreversibility of heat transfer across a finite temperature difference (thermal resistance) and the irreversibility of fluid flow across a finite pressure drop (fluid flow resistance) was exploited by Bejan [30] and Radcenco [31], and was further studied by Bejan [32–34] and Chen et al. [35,36]. For the open Brayton cycles, the principle of optimally tuning the air flow rate and subsequent distribution of pressure drops was used [32–36]. The further step of this paper beyond Refs. [5–8,26–29] is to analyze and optimize the performance of the combined Brayton and inverse Brayton cycles [7] with considerations of the 11 pres-

k h

q s w

excess air ratio adiabatic temperature ratio density (kg/m3) temperature ratio of the top cycle relative pressure drop

Subscripts a air c compressor cc compressor–combustor cf combustor ct combustor-turbine e environment g gas i inlet m mean max maximum min minimum opt optimal p pressure s entropy t turbine v volume 0 ambient 1; 2; 3; . . . ; 10 state points in the cycle/sequence numbers

sure drop losses in the intake, low-pressure compression, highpressure compression, combustion, expansion and discharge processes and flow process in the piping, the heat transfer loss to ambient, the irreversible compression and expansion losses in the compressors and the turbines, and the irreversible combustion loss in the combustion chamber by using the similar principle and method used in Refs. [26–29]. The thermodynamic performance is optimized by adjusting the bottom cycle pressure ratio, the mass flow rate and the distribution of pressure losses along the flow path. Also, it is shown that the power output has a maximum with respect to the bottom cycle pressure ratio, the air mass flow rate or any of the overall pressure drops, and the maximized power output has an additional maximum with respect to the top cycle pressure ratio. When the optimization is performed with the constraints of a fixed fuel flow rate and the power plant size, the power output and efficiency can be maximized again by properly allocating the fixed overall flow area among the compressor inlet of the top cycle and the compressor outlet of the bottom cycle. The numerical examples show the effects of design parameters on the power output and heat conversion efficiency.

2. Physical model The proposed system shown in Fig. 1 is constructed from two cycles [7]. A Brayton cycle is the top cycle and is used as a gas generator to power the bottom cycle. The purpose of the turbine in the top cycle is solely to power the compressor, the power produced by the complete system is delivered via the bottom cycle. The bottom cycle begins with an expansion process, followed by cooling at constant pressure and a compression processes that delivers the working fluid at the required stack condition. This part is known as an inversed Brayton cycle and is used to augment power production by expanding further the gas turbine exhaust gas beyond atmospheric conditions.

W. Zhang et al. / Applied Thermal Engineering 29 (2009) 2885–2894

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(f) In view of the ideal gas model, the temperature increase experienced by the gas through the combustion chamber is equivalent to the isobaric process 20 –30 at pressure P 0 ¼ P  DPcc . A fraction (Q_ cf ) of the heating rate produced by the burning fuel (Q_ f ) leaks directly into the ambient

Fig. 1. System layout [7].

The plant model is expressed using pressure drop and mass flow rate distributions and temperature–entropy diagram, as shown in Fig. 2. There are six components, the high-pressure compressor, the combustion chamber, the high-pressure gas turbine, the low-pressure gas turbine, the heat exchanger and low-pressure compressor. The feature of flow through each component is that there is a pressure drop that increases with the flow rate. The key of this study is to determine the optimal bottom cycle pressure ratio and the optimal flow rate (or optimal pressure drops) that allow the power plant to operate at maximum power output or maximum thermal conversion efficiency. The similar studies for a simple fluid flow power converters were performed by Bejan [32–34] and Chen et al. [35,36]. Proceeding along the path followed by the working fluid, the following modeling assumptions are made: (a) The working fluid (air and air + products of combustion) is an ideal gas with a specific heat that depends on temperature and composition. (b) The air flows into the compressor of the top cycle (process 0–1) irreversibly and accompanied by the pressure drop DP1 = P0  P1 and the entropy increase Ds1 at the ambient temperature T0 [31]. In the following analysis, P represents overall pressure. (c) The air compression process 1–2 of the top cycle is adiabatic and irreversible, leading to the entropy increase Dsc1. In Fig. 2b this process is represented schematically by the isentropic compression 1–2s0 followed by the throttling process 2s0 –2, which accounts for the pressure drop DPc1 associated with fluid friction through the compressor stages of the top cycle. (d) The combustion process and flow through the combustion chamber are characterized by the pressure drop DPcc due to the flow into the combustion chamber (process 2–30 ) and the entropy increase Dscc. (e) The pressure drop associated with the flow out of the combustion chamber and into the turbine (process 30 –3) is DPct. The process is accompanied by the entropy increase Dsct.

through the walls of the combustion chamber [30,31,33]. (g) The turbine expansion process 3–4 of the top cycle is modeled as adiabatic and irreversible with the entropy increase Dst1. This process is equivalent to the isentropic expansion 3–4s0 from P3 ¼ P0  DPct to P4s0 ¼ P 4 þ DP t1 , followed by the adiabatic throttling process 4s0 –4 accounting for the pressure drop DPt1 through the blades and vanes of the turbine of the top cycle. (h) The pressure drop associated with the flow out of the turbine of the top cycle and into the turbine of the bottom cycle (process 4–40 ) is DPtt. The process is accompanied by the entropy increase Dstt. The turbine expansion process 40 –5 is modeled as adiabatic and irreversible with the entropy increase Dst2. This process is equivalent to the isentropic expansion 40 –5s0 from P04 ¼ P 4  DP tt to P5s0 ¼ P 5 þ DP t2 , followed by the adiabatic throttling process 5s0 –5 accounting for the pressure drop DPt2 through the blades and vanes of the turbine of the bottom cycle. (i) The air flow through the heat exchanger (process 5–60 ) is characterized by the overall pressure drop DPci. This process is represented schematically by the throttling process 5–50 followed by the isobaric process 50 –60 at pressure P 50 , which accounts for the pressure drop DPci associated with fluid friction through the heat exchanger. The effectiveness of the heat exchanger is defined as e = (T5  T6)/(T5  T0). (j) The air flows into the low-pressure compressor (process 60 – 6) of the bottom cycle irreversibly and accompanied by the pressure drop DP2 and the entropy increase Ds2 at the ambient temperature T0 [31]. (k) The air compression process 6–7 of the bottom cycle is adiabatic and irreversible, leading to the entropy increase Dsc2. In Fig. 2b this process is represented schematically by the isentropic compression 6–7s0 followed by the throttling process 7s0 –7, which accounts for the pressure drop DPc2 associated with fluid friction through the compressor stages of the bottom cycle. (l) The discharge of the gas stream from the compressor of the bottom cycle (process 7  e) causes another pressure drop DPe = P7  P0 and entropy increase Dse at temperature Te.

3. Cycle analysis There are 11 flow resistances encountered by the gas stream for the combined Brayton and inverse Brayton cycles. Four of these, the friction through the blades and vanes of the compressors and the turbines, are related to the isentropic efficiencies gc1, gc2, gt1 and gt2, respectively. In principle, these resistances can be rendered negligible by minimizing friction in the compressors and turbines in the limit (gc1, gc2, gt1, gt2) ? 1. The remaining seven flow resistances, however, are always present because of the changes in flow cross-section at the compressor inlet of the top cycle, combustion inlet and outlet, turbine outlet of the top cycle, turbine outlet of the bottom cycle, heat exchanger inlet, and compressor inlet of the bottom cycle. These resis_ _ and the net power output W tances control the air flow rate m [26–36]. For example, the pressure drop at the compressor inlet of the top cycle is given by

DP 1 ¼ K 1

  1 q0 m21 2

ð1Þ

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Fig. 2. Combined Brayton and inverse Brayton cycles plant model of Fig. 1: (a) pressure drop and mass flow rate distributions and (b) temperature–entropy diagram and the flow resistances.

where K1 is the contraction pressure loss coefficient, and v1 is the average air velocity through the inlet flow cross-section A1, see Fig. 2a. It is assumed that the flow is highly turbulent and, as a first approximation, K1 is a constant when the change in the flow crosssection is fixed [37]. The air mass flow rate through the same cross_ ¼ A1 q0 m1 , or section is m

_ ¼ A1 m



2 q P0 K1 0

1=2

1=2

w1

ð2Þ

where w1 = DP1/P0 is the relative pressure drop associated with the first flow resistance. The modeling of the flow through the compressor stages of the top cycle continues with the apparent compressor pressure ratio b1 = P2/P0 as an input parameter [38]. The effective pressure ratio bc1 = P2/P1 = b1/(1  w1) is related to the isentropic temperature raðc 1Þ=ca1 , where tio (hc1s) across the compressor, hc1s ¼ T 2s =T 1 ¼ bc1a1 the ratio of the air specific heats ca1 ¼ ðC p =C v Þair decreases as the mean air temperature Tma increases, ca1 ¼ 1:438

W. Zhang et al. / Applied Thermal Engineering 29 (2009) 2885–2894

ð1:05  104 Þ  T ma . The empirical correlation for ca1 was developed by Radcenco [39], and is valid with 0.5% in the range 350 K < Tma < 1000 K, where Tma = T0(1 + hc1s)/2. The specific work required by the low-pressure compressor, 1 wc1 ¼ g1 c1 ðh2s  h1 Þ ¼ gc1 ca1 RT 0 ðhc1s  1Þ=ðca1  1Þ, can be related to the pressure drop through the blades and vanes by writing hc1 ¼ T 2 =T 1 ¼ 1 þ ðhc1s  1Þ=gc1 , and noting that h2s0 ¼ h2 and Taking the constant wc1 ¼ DP c1 =P 2 ¼ ðhc1 =hc1s Þca1 =ðca1 1Þ  1. A1 ð2=K 1 Þ1=2 P0 ðRT 0 Þ1=2 whose unit is the same as that of the energy interaction as the denominator [27], the resulting dimensionless _ c1 ¼ mw _ c1 of the expression for the compressor power input W top cycle is

_ c1 ¼ W

W c1 A1 ð2=K 1 Þ1=2 P0 ðRT 0 Þ1=2

¼

ca1 ðhc1s  1Þ 1=2 w gc1 ðca1  1Þ 1

ð3Þ

The relative pressure drop associated with the flow of compressed air into the combustion chamber is dictated by mass con_ ¼ A1 q0 m1 ¼ A2 q2 m2 , where A2 and v2 are the servation, m compressor outlet flow cross-sectional area and the mean velocity based on A2 of the top cycle. The result is

wcc ¼

DPcc hc1 w1 ¼ 2 P2 b1 ðA2 =A1 Þ2 ðK 1 =K 2 Þ

ð4Þ

In accordance with assumption (d), the heat leak from the combustor to the ambient is accounted for in terms of combustor efficiency

_ Q_ f ¼ 1  Q_ cf =Q_ f gcf ¼ Q=

ð5Þ

The heat transfer rate received by the gas stream is _ g cpg ðT 3  T 2 Þ, where m _ g is the gas mass flow rate, Q_ ¼ gcf Q_ f ¼ m _ þm _f ¼m _ f ðkL0 þ 1Þ, k and L0 are the excess air ratio and _g ¼m m theoretical air quantity

gcf 1 cgc  1 Q f k¼   1 L0 cgc RT 0 s  hc1

! ð6Þ

s ¼ T max =T 0

ð7Þ

where s is the overall temperature ratio and Q f ¼ Q_ f =mf . The fuel considered in this study is kerosene with a composition by weight of 86.08% carbon and 13.92% hydrogen, theoretical air L0 = 14.64(kg air)/(kg fuel), and Qf = 43.1  103 kJ/(kg fuel) [40]. The ratio of specific heats of the gas in the combustor, cgc = (Cp/ Cv)gas, has been correlated [39] as a function of k and a average gas temperature Tmgc = T0(hc1 + s)/2

cgc ¼ 1:254 

0:0372 76:7 þ k T mgc

1=2

¼

W t1 ¼

Q f w1 kL0 RT 0

A1 ð2=K 1 Þ P0 ðRT 0 Þ   1 cgc ðs  hc1 Þ 1=2 ¼ 1þ w kL0 ðcgc  1Þgcf 1

ð9Þ

The corresponding heat transfer received by the gas stream is

  1 cgc ðs  hc1 Þ 1=2 w Q ¼ gcf Q f ¼ 1 þ ðcgc  1Þ 1 kL0

ð10Þ

The relative pressure drop associated with the flow out of the combustor and into the turbine inlet of the top cycle (cross-sectional area A3) is determined from mass conservation _ þ 1=ðkL0 Þ ¼ A1 q0 m1 ½1 þ 1=ðkL0 Þ ¼ A3 q3 m3 , with K3 trea_ g ¼ m½1 m ted as a constant. The result is

ð11Þ

_ t1 W

A1 ð2=K 1 Þ1=2 P 0 ðRT 0 Þ1=2   1=2 1 gt1 sð1  1=ht1s Þcg1 w1 ¼ 1þ kL0 cg1  1

ð12Þ

Because of the turbine of the top cycle utilized to drive the compressor of the top cycle, i.e. W t1 ¼ W c1 , it can be obtained

  1 gt1 sð1  1=ht1s Þcg1 ca1 ðhc1s  1Þ 1þ ¼ kL0 cg1  1 gc1 ðca1  1Þ

ð13Þ

The pressure drop associated with the flow into the turbine of the bottom cycle is DPtt ¼ K 4 q4 m24 =2 (or DP tt ¼ K 40 q40 m240 =2), where K4 (K 40 ) and v4 (v 40 ) are the turbine of the bottom cycle inlet flow cross-section area and the mean velocity based A4(A40 ), taking DPtt ¼ K 4 q4 m24 =2 for calculation. The relative pressure drop associated with the flow out of the turbine of the top cycle and into the turbine of the bottom cycle is deter_ þ 1=ðkL0 Þ ¼ A1 q0 _ g ¼ m½1 mined from mass conservation m m1 ½1 þ 1=ðkL0 Þ ¼ A4 q4 m4 , with K4 treated as a constant. The result is

wtt ¼

1=2

_

Qf 1=2

Dpct ½1 þ 1=ðkL0 Þ2 sw1 0 ¼ P ðK 1 =K 3 ÞðA3 =A1 Þ2 ð1  wcc Þ2 b21

The overall pressure drop associated with the flow of compressed working fluid through the combustion chamber is DPcf = DPcc + DPct = P2wcc + (P2  DPcc)wct, and the overall relative pressure drop is wcf = wcc + (1  wcc)wct ffi wcc + wct. The common range of wcf is 0.05 6 wcf(=DPcf/P) 6 0.07. The modeling of the flow through the turbine of the top cycle continues with the apparent turbine pressure ratio b2 ¼ P0 =P 4 as an input parameter. The effective pressure ratio bt1 = P3/ P4 = b2(1  wct) is related to the isentropic temperature ratio ht1s ðc 1Þ=cg1 across the turbine of the top cycle, ht1s ¼ T 3 =T 4s ¼ bt1g1 , where the ratio of the gas specific heats cg1 in the temperature range occupied by the turbine of the top cycle is correlated by the same Eq. (8) where Tmg1 is the average temperature Tmg1 = sT0(1 + 1/ht1s)/ 2. The specific power output of the turbine of the top cycle is wt1 = gt1RT0s(1  1/ht1s)cg1/(cg1  1), where the isentropic efficiency gt1 is related to the pressure drop associated with the friction through the turbine blades and vanes, wt1 = DPt1/P4. Take noting of ht1 ¼ T 3 =T 4 ¼ 1=ð1  gt1 þ gt1 =ht1s Þ and h4s0 ¼ h4 (Fig. 2b), one can get wt1 ¼ ðht1s =ht1 Þcg1 =ðcg1 1Þ  1, where ht1 is a function of gt1. In conclusion, the turbine power output of the _ t1 ¼ m _ g wt1 can be expressed in dimensionless form as top cycle W

ð8Þ

The excess air k is calculated iteratively between Eqs. (6) and (8). It can be shown that the heat transfer produced by the burning fuel can be nondimensionalized and expressed as follows:

Qf ¼

wct ¼

2889

 2 DPtt 1 b22 sw1 ¼ 1þ  kL0 P4 ðK 1 =K 4 ÞðA4 =A1 Þ2 ht1 b21 ð1  wcc Þ2

ð14Þ

The modeling of the flow through the turbine of the bottom cycle continues with the apparent turbine pressure ratio b3 = P4/P5 as an input parameter. The effective pressure ratio bt2 ¼ P40 P5 ¼ b3 ð1  wtt Þ is related to the isentropic temperature ratio ht2s across ðc 1Þ=cg2 , where the turbine of the bottom cycle, ht2s ¼ T 40 =T 5s ¼ bt2g2 the ratio of the gas specific heats cg2 in the temperature range occupied by the turbine of the bottom cycle is correlated by the same Eq. (8) where Tmg2 is the average temperature Tmg2 = sT0(1 + 1/ht2s)/2/ht1. The specific power output of the turbine of the bottom cycle is wt2 = gt2RT0s(1  1/ht2s)cg2/(cg2  1)/ht1, where the isentropic efficiency gt2 is related to the pressure drop associated with the friction through the turbine blades and vanes, wt2 = DPt2/P5. Take noting of ht2 ¼ T 40 =T 5 ¼ 1=ð1  gt2 þ gt2 =ht2s Þ and h5s0 ¼ h5 (Fig. 2b), one can get wt2 ¼ ðht2s =ht2 Þcg2 =ðcg2 1Þ  1, where ht2 is a function of gt2. In conclusion, the turbine power out_ t2 ¼ m _ g wt2 can be expressed in dimenput of the bottom cycle W sionless form as

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W. Zhang et al. / Applied Thermal Engineering 29 (2009) 2885–2894

_ t2 W

W t2 ¼

A1 ð2=K 1 Þ1=2 P0 ðRT 0 Þ1=2   1=2 1 gt2 sð1  1=ht2s Þcg2 w1 ¼ 1þ kL0 ðcg2  1Þht1

ð15Þ

The relative pressure drop associated with the flow out the turbine and into the heat exchanger (cross-sectional area A5) of the _ þ 1= _ g ¼ m½1 bottom cycle is dictated by mass conservation, m ðkL0 Þ ¼ A1 q0 m1 ½1 þ 1=ðkL0 Þ ¼ A5 q5 m5 , where A5 and v5 are the turbine of the bottom cycle outlet flow cross-sectional area and the mean velocity based on A5. The result is

 2 DPci 1 b22 b23 sw1 wci ¼ ¼ 1þ  2 kL0 P5 ðA5 =A1 Þ ðK 1 =K 5 Þð1  wcc Þ2 b21 ht1 ht2

ð16Þ

where the pressure loss coefficient K5 is defined by DP ci ¼ K 5 q5 m25 =2 and treated as a constant. The heat transfer rate in heat exchanger is

_ g ðh5  h6 Þ Q_ i ¼ m _ þ 1=ðkL0 Þcai sRT 0 ð1  1=hi Þ=ðcai  1Þ=ht1 =ht2 ¼ m½1

ð17Þ

where hi ¼ T 50 =T 60 ¼ 1=ð1  e þ eht1 ht2 =sÞ is the inlet and outlet temperature ratio of the working fluid through the heat exchanger, and cai in the temperature range occupied by the heat exchanger is correlated by the same Eq. (8), where Tmai = T0s(1 + 1/hi)/ht1/ht2/2. Q_ i can be nondimensionalized and expressed as follows:

Q_ i

Qi ¼

1=2

A1 ð2=K 1 Þ P0 ðRT 0 Þ1=2    1 1 cai s 1=2 ¼ 1þ w 1 kL0 hi cai  1 ht1 ht2 1

ð18Þ

The relative pressure drop associated with the flow out the heat exchanger and into the compressor of the bottom cycle is dictated _ þ 1=ðkL0 Þ ¼ A1 q0 m1 ½1 þ 1= _ g ¼ m½1 by mass conservation, m ðkL0 Þ ¼ A6 q6 m6 , where A6 and v6 are the compressor inlet flow cross-sectional area and the mean velocity based on A6 of the bottom cycle. The result is

 w2 ¼

1þ

1 kL0

2 

sb2 b3 bi w1 ðA6 =A1 Þ2 ðK 1 =K 6 Þð1  wci Þð1  wcc Þb1 hi ht1 ht2 ð19Þ

where bi (bi = P0/P6) is the ratio of the ambient pressure to the pressure of the compressor inlet of the bottom cycle. The modeling of the flow through the compressor stages of the bottom cycle continues with the apparent compressor pressure ratio b4 ¼ P 7 =P 50 as an input parameter. The effective pressure ratio bc2 = P7/P6 = b4/(1  w2) is related to the isentropic temperature ratio (hc2s) across the compressor of the bottom cycle, ðc 1Þ=ca2 , where ca2 in the temperature range occuhc2s ¼ T 7s =T 6 ¼ bc2a2 pied by the compressor of the bottom cycle is correlated by the same Eq. (8), where Tma2 = T0s(1 + hc2s)/hi/ht1/ht2/2. Take noting of h7s0 ¼ h7 ; wc2 ¼ DP c2 =P 7 ¼ ðhc2 =hc2s Þca2 =ðca2 1Þ  1 and hc2 = T7/T6 = 1 1 + (hc2s  1)/gc2, one can get wc2 ¼ ðh7s  h6 Þ=gc2 ¼ g1 c2 hi 1 h1 h c RT s ðh  1Þ=ð c  1Þ. The resulting dimensionless 0 c2s a2 t1 t2 a2 _ c2 ¼ m _ g wc2 is expression for the compressor power input W

W c2 ¼

_ c2 W

A1 ð2=K 1 Þ1=2 P0 ðRT 0 Þ1=2   1 ca2 sðhc2s  1Þ ¼ 1þ w1=2 kL0 gc2 hi ht1 ht2 ðca2  1Þ 1

ð20Þ

The analysis of the compression through the compressor of the bottom cycle begins with a guess for the relative pressure drop at

the compressor outlet, w7 = DPe/P0, where P0 is the known ambient pressure, DPe ¼ K 7 q7 m27 =2, K7 is pressure loss coefficient, and v7 is the mean velocity based on the flow cross-section A7. The exit pressure drop w7 will be calculated subsequently by trial and error and _ g ¼ m½1 _ þ 1=ðkL0 Þ ¼ the exit pressure is w7 = DPe/P0. Noting m A1 q0 m1 ½1 þ 1=ðkL0 Þ ¼ A7 q7 v 7 (according to mass continuity), after some algebra, one has

ð1 þ w7 Þw7 ffi w7 ¼

ð1 þ 1=kL0 Þ2 shc2 ðA7 =A1 Þ2 ðK 1 =K 7 Þhi ht1 ht2

w1

ð21Þ

The cooling rate experienced by the exhaust as it reaches the _ g C Pg0 ðT e  T 0 Þ, or in dimensionambient temperature T0 is Q_ 0 ¼ m less form

_

Q0 1=2

Q0 ¼

A1 ð2=K 1 Þ P0 ðRT 0 Þ1=2   1=2 1 cg0 ðshc2 =hi =ht1 =ht2  1Þw1 ¼ 1þ kL0 cg0  1

ð22Þ

where cg0 is evaluated based on Eq. (8) with Tmg0 = T0(shc2/hi/ht1/ ht2 + 1)/2. To summarize the analytical formulae of the model, one notes the expressions for compressor input power Eq. (3) of the top cycle, heating produced by the fuel Eq. (9), turbine power output Eq. (12) of the top cycle, turbine power output Eq. (15) of the bottom cycle, heat loss of the heat exchanger Eq. (18), compressor input power Eq. (20) of the bottom cycle, and heat rejection due to exhaust 1=2 Eq. (22). Each of these quantities is proportional to w1 (or to any one of the other 10 pressure drops), which in turn is propor_ g. _ m _ f or m tional to the flow rate, m; _ t1 ; W _ t2 ; Q_ i ; _ c1 ; Q_ f ; W Therefore, the energy interactions W _ c2 and Q_ 0 decrease in proportion with the flow rate, for example, W the net power output is zero when the flow rate is zero. When the energy Q_ f released by fuel inside the combustion chamber is taken as a constraint, the maximum power design is also the most economical one, i.e. the one with maximum efficiency and, at the same time, minimum total entropy generation rate. And the net power output per mass flow rate decreases with the increase in the 11 resistances (or the 11 pressure drops) along the flow path of the working fluid, which indicates that there should be an optimal flow rate (or any one of the 11 pressure drops) which maximizes the net power output and in turn is proportional to the mass flow rate. The overall energy balance for the power plant indicates that _ c2 ¼ Q_ f  Q_ cf  Q_ i  Q_ 0 is the net power output. The _ ¼W _ t2  W W first law efficiency of the combined cycle power plant is

_ W

W

_ W

g1 ¼ _ ¼ ¼ gcf _ Qf Qf Q

ð23Þ

1=2 _ where W ¼ W=½A P0 ðRT 0 Þ1=2  is the dimensionless net 1 ð2=K 1 Þ _ Q_ is the thermal conversion efficiency g of power output and W= the cycle supplied with Q_ ,

_ c sð1  h1 cg0 ðcgc  1Þðshc2 =hi =ht1 =ht2  1Þ W i Þðcgc  1Þ  g ¼ _ ¼ 1  ai cgc ht1 ht2 ðs  hc1 Þðcai  1Þ cgc ðcg0  1Þðs  hc1 Þ Q ð24Þ

The objective of this study is to solve @W=@w1 ¼ 0 and @W=@bi ¼ 0 numerically, and to determine the optimal fuel flow rate and pressure drops that maximize the net power output. 4. Optimal pressure drop and fuel flow rate for maximum power The effects of the bottom cycle pressure ratio, the air mass flow rate and pressure drops on the net power output are examined by

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Fig. 3. The W—w1 ; g—w1 ; Q —w1 ; Q i —w1 and Q 0 —w1 characteristics.

Fig. 5. The W max —b1 ; ðbiopt ÞW —b1 and ðw1opt Þw —b1 characteristics.

using numerical examples. The range covered by the calculations is 0 6 w1 6 0.6, 5 6 b1 6 40, 1 6 bi 6 2.5, 4 6 s 6 6, P0 = 0.1013 MPa, gc1 = 0.9, gc2 = 0.87, gt1 = 0.85, gt2 = 0.83 and gcf = 0.99. The ratio of the outermost equivalent flow cross-sections (compressor inlet of the top cycle/turbine outlet of the bottom cycle) covered the range 0.25 6 a1–7 6 4, where a1–7 is the dimensionless group

In the calculations, a1–3 = 1/2, a1–2 = a1–4 = a1–5 = a1–6 = a1–7 = 1/ 3, and T0 = 300 K are set. Figs. 3 and 4 illustrate the behaviors of the main dimensionless quantities of interest (W; g; Q 0 ; Q ; Q i ) as w1 and bi increases, respectively. From which, one can see that the thermal efficiency g decreases with the increase in the compressor inlet relative pressure drop w1 of the top cycle, the dimensionless power output W reaches its maximum value at the optimal value w1.opt, and the dimensionless power output W and the thermal efficiency g reach their maximum values at the optimal values of the compressor pressure ratio bi of the bottom cycle, respectively. Fig. 5 shows the maximum dimensionless power output W max of the combined cycle, the corresponding optimal compressor pressure ratio (biopt)W of the bottom cycle and the optimal compressor inlet relative pressure drop (w1opt)W of the top cycle versus the compressor pressure ratio b1. From which, one can see that the maximum dimensionless power output W max reaches its maxima (W max;2 ) at the optimal compressor pressure ratio (b1opt)W of the

top cycle. The optimal compressor pressure ratio (biopt)W of the bottom cycle decreases with the increase in the compressor pressure ratio b1 of the top cycle. The optimal compressor inlet relative pressure drop (w1opt)W of the top cycle decreases with the increase in the compressor pressure ratio b1 of the top cycle. Figs. 6–9 show the effects of a1–7 on the cycle twice maximum dimensionless power output W max;2 , corresponding optimal compressor inlet relative pressure drop (w1opt,2)W and pressure ratio (b1opt)W of the top cycle and corresponding optimal compressor inlet pressure (P6opt,2)W of the bottom cycle versus the temperature ratio s of the top cycle and the effectiveness e of the heat exchanger, respectively. From which, one can see that W max;2 and (w1opt,2)W increase with the increases in s and e, and the decrease in a1–7. Both (b1opt)W and (P6opt,2)W increase with the increases in a1–7 and s and the decrease in e. Fig. 10 shows the excess air ratio k versus the compressor inlet relative pressure drop w1 of the top cycle. It illustrates that k increases with the increase in w1. An important feature of the maximum power condition identified in the preceding section is that the thermal conversion efficiency g at w1 = (w1)opt is less than the efficiency at w1 less than (w1)opt. In fact, g is at maximum when w1 = 0, i.e. when the entropy generated by the 10 flow resistance is zero. Fig. 5 shows that the optimal pressure ratio of the top cycle corresponding to the maximum power output is (b1opt)W = 13. If the energy Q_ f released by fuel inside the combustion chamber is taken as the constraint, it is shown that the maximum power design is

Fig. 4. The W—bi ; g—bi ; Q —bi ; Q i —bi and Q 0 —bi characteristics.

Fig. 6. The effect of a1–7 on the W max :2 —s and ðw1opt:2 Þw —s characteristics.

a1—7 ¼

 1=2 A1 K 7 ; A7 K 1

a1i ¼

 1=2 A1 K i Ai K 1

ði ¼ 2; 3; 4; 5; 6; 7Þ

ð25Þ

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Fig. 7. The effect of a1–7 on the ðb1opt Þw —s and ðP 6opt:2 Þw —s characteristics.

Fig. 10. The k  w1 characteristics.

5. Effects of fuel and size constraints For a more practical look at what ‘‘optimal” operation means, the operation of the preceding model subject to two practical constraints is considered here. The first is that the energy Q_ f released by fuel inside the combustion chamber. It is shown that in this case, the maximum power design is also the most economical one, i.e. the one with maximum efficiency and, at the same time, minimum total entropy generation rate. The Q_ f constraint can be expressed analytically as 1=2

Q f w1 ¼ constant Q_ f ¼ A1 ð2=K 1 Þ1=2 P0 ðRT 0 Þ1=2 kL0 RT 0

Fig. 8. The effect of a1–7 on the W max :2 —e and ðw1opt:2 Þw —e characteristics.

ð26Þ

The second constraint refers to the total size of the combined cycle power plant. Related to the overall size and weight is the sum of the compressor inlet and turbine outlet flow areas, A1 + A6 and A4 + A5. Instead of fixing the sum (A1 + A6 + A4 + A5), it is convenient to include in the constraint the pressure loss coefficients,

A1 K 1=2 1

þ

A5 K 1=2 5

¼ A ¼ constant

ð27Þ

and seek the optimal flow area allocation ratio x defined by ¼ xA and A5 =K 1=2 ¼ ð1  xÞA . Combining Eq. (26) with Eq. A1 =K 1=2 5 1 (27) yields:

Qf ¼

Q_ f A P0 ðRT 0 Þ1=2

x ¼ C w1=2 ¼ constant k 1

ð28Þ

The constant C = 21/2Qf/(RT0L0) is dictated by the choice of fuel and ambient absolute temperature. Eq. (16) becomes

 2 1 x2 b22 b23 sw1 wci ¼ 1 þ  kL0 ð1  wcc Þ2 ð1  xÞ2 b21 ht1 ht2

ð29Þ

therefore

   pffiffiffi 2cai xs 1=2 1 1 ¼ 1 þ w1 ð30Þ 1  1=2 kL h c 0 i A P0 ðRT 0 Þ ai  1 ht1 ht2 p ffiffiffi   2cg0 ðshc2 =hi =ht1 =ht2  1Þxw11=2 1 Q_ 0 Q 0 ¼ ¼ 1 þ ð31Þ kL0 cg0  1 A P0 ðRT 0 Þ1=2 Q_ i

Q i ¼ Fig. 9. The effect of a1–7 on the ðb1opt Þw —e and ðP 6opt:2 Þw —e characteristics.

also the most economical one, i.e. the one with maximum efficiency and, at the same time, minimum total entropy generation rate. However, if the total size of the combined cycle power plant is taken as the constraint, it can be seen from the following section that the optimal pressure ratio of the top cycle corresponding to the maximum efficiency is (b1opt)W > 30. In order to obtain the trade-off performance between the power output and efficiency, b1 ¼ 16 is chosen in the analysis.

and the first law efficiency (Eq. (23)) becomes

g1 ¼ 1 

Q i



Q 0

¼1

pffiffiffi 2kcai sð1 þ 1=ðkL0 ÞÞð1  h1 i Þ Cht1 ht2 ðcai  1Þ

Qf Qf pffiffiffi 2kcg0 ðshc2 =hi =ht1 =ht2  1Þð1 þ 1=ðkL0 ÞÞ  Cðcg0  1Þ

ð32Þ

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Fig. 13. The g1 max —Q f  ; xopt —Q f  ; ðw1opt Þg —Q f  and ðP 6opt Þg —Q f  characteristics. Fig. 11. The g1–bi, g1–w1 and g1–x characteristics.

Fig. 11 shows the characteristic of the power plant efficiency g1 versus the compressor pressure ratio bi of the bottom cycle, the compressor inlet relative pressure drop w1 of the top cycle and the area allocation ratio x. It indicates that there exist an optimal compressor pressure ratio biopt of the bottom cycle, an optimal compressor inlet relative pressure drop w1opt of the top cycle and an optimal area allocation ratio xopt that maximize the combined cycle power plant efficiency, i.e. (g1)max. The numerical maxima (g1)max can be obtained through simultaneous solution of the following equations:

@ g1 ¼ 0; @w1

@ g1 ¼0 @bi

ð33Þ

Fig. 12 shows the cycle maximum first law efficiency g1max, the corresponding optimal compressor inlet relative pressure drop (w1opt)g of the top cycle, the corresponding optimal area allocation ratio xopt and the corresponding optimal compressor inlet pressure (P6opt)g of the bottom cycle versus the compressor pressure ratio with. From which, one can see that g1max reaches its maxima (g1max,2) at optimal compressor pressure ratio (b1opt)g of the top cycle. Both xopt and (w1opt)g increase with the increase in b1. (P6opt)g increases with the increase in b1.

Fig. 12. The g1max–b1, xopt–b1, (w1opt)g–b1 and (P6opt)g–b1 characteristics.

Fig. 13 shows g1max, (P6opt)g, (w1opt)g and xopt versus Q f  . It illustrates that g1max decreases with the increase in Q f  ; (P6opt)g, (w1opt)g and xopt increase with the increase in Q f  . 6. Conclusion Based on the work of Refs. [5–8,26–29], this paper extends the finite time thermodynamic theory and method to the performance analysis and optimization of the combined Brayton and inverse Brayton cycles power plant. The analytical formulae about the relation between power output and the compressor pressure ratio of the top cycle are derived with the 11 pressure drop losses in the intake, compression, combustion, expansion and flow process in the piping, the heat transfer loss to ambient, the irreversible compression and expansion losses in the compressors and the turbines, and the irreversible combustion loss in the combustion chamber. The performance is optimized by adjusting the compressor pressure ratio of the bottom cycle, the air mass flow rate and the distribution of pressure losses along the flow path. The main conclusions reached are: (1) The net power output has a maximum with respect to the air mass flow rate or any one of the 11 pressure drops of the cycle, the compressor pressure ratio of the bottom cycle, and the maximized net power output has an additional maximum with respect to the compressor pressure ratio of the top cycle, Figs. 3–5. (2) When the thermodynamic optimization is conducted by taking the energy released by fuel inside the combustion chamber as a constraint, the maximum power design is also the most economical one, i.e. the one with maximum efficiency and, at the same time, minimum total entropy generation rate. (3) If the total size of the combined cycle power plant is taken as a constraint, there exists another optimal compressor pressure ratio of the top cycle corresponding to the maximum efficiency. The thermal efficiency is maximized by optimizing the compressor pressure ratios of the bottom cycles and the compressor inlet pressure drop of the top cycle. (4) When the optimization is conducted subject to fixed fuel flow rate and total size, there is a trade-off in how the size should be allocated among components. With respect to the total flow cross-section area constraint, there exists an optimal flow area allocation ratio which maximizes the thermal efficiency.

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The results of this paper can provide new theoretical guidance for the performance improvement of practical open-cycle gas turbine power plant, which are and/or will be widely used in industry, warship, aircraft, train, space power station, etc. Acknowledgements This paper is supported by Program for New Century Excellent Talents in University of PR China (Project No. NCET-04-1006) and The Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Project No. 200136). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript. References [1] D.L. Chase, Combined Cycle Development Evolution and Future. GE Power Systems, GER-4206, 2001. [2] T.H. Frost, A. Anderson, B. Agnew, A hybrid gas turbine cycle (Brayton/ Ericsson): an alternative to conventional combined gas and steam turbine power plant, Proc. IMechE, Part A: J. Power Energy 211 (A2) (1997) 121–131. [3] J. Zheng, F. Sun, L. Chen, C. Wu, Exergy analysis for a Braysson cycle, Exergy, An. Int. J. 1 (1) (2001) 41–45. [4] T. Zheng, L. Chen, F. Sun, C. Wu, Power, power density and efficiency optimization of an endoreversible Brayton cycle, Exergy, An. Int. J. 2 (4) (2002) 380–386. [5] S. Fujii, K. Kaneko, K. Otani, Y. Tsujikawa, Mirror gas turbine: a newly proposed method of exhaust heat recovery, Trans. ASME, J. Eng. Gas Turb. Power 123 (3) (2001) 481–486. [6] M. Bianchi, G. Negri di Montenegro, A. Peretto, Inverted Brayton cycle employment for low temperature cogeneration applications, Trans. ASME, J. Eng. Gas Turb. Power 124 (3) (2002) 561–565. [7] B. Agnew, A. Anderson, I. Potts, T.H. Frost, M.A. Alabdoadaim, Simulation of combined Brayton and inverse Brayton cycles, Appl. Therm. Eng. 23 (8) (2003) 953–963. [8] W. Zhang, L. Chen, F. Sun, C. Wu, Second-law analysis and optimization for combined Brayton and inverse Brayton cycles, Int. J. Ambient Energy 28 (1) (2007) 15–26. [9] M.A. Alabdoadaim, B. Agnew, A. Alaktiwi, Examination of the performance envelope of combined Rankine, Brayton and two parallel inverse Brayton cycles, Proc. IMechE Part A: J. Power Energy 218 (A6) (2004) 377–386. [10] M.A. Alabdoadaim, B. Agnew, I. Potts, Examination of the performance of an unconventional combination of Rankine, Brayton and inverse Brayton cycles, Proc. IMechE, Part A: J. Power Energy 220 (A4) (2006) 305–313. [11] M.A. Alabdoadaim, B. Agnew, I. Potts, Performance analysis of combined Brayton and inverse Brayton cycles and developed configurations, Appl. Therm. Eng. 26 (14–15) (2006) 1448–1454. [12] B. Andresen, Finite-time Thermodynamics, Physics Laboratory II, University of Copenhagen, 1983. [13] S. Sieniutycz, P. Salamon (Eds.), Advances in Thermodynamics, Finite Time Thermodynamics and Thermoeconomics, vol. 4, Taylor & Francis, New York, 1990. [14] A. De Vos, Endoreversible Thermodynamics of Solar Energy Conversion, Oxford University, Oxford, 1992.

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