International Journal of Sustainable Energy

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International Journal of Sustainable Energy

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Power optimization of a regenerated closed variable-temperature heat reservoir Brayton cycle

To cite this Article: Chen, Lingen, Sun, Fengrui and Wu, Chih , 'Power optimization of a regenerated closed variable-temperature heat reservoir Brayton cycle', International Journal of Sustainable Energy, 26:1, 1 - 17 To link to this article: DOI: 10.1080/14786450701259416 URL: http://dx.doi.org/10.1080/14786450701259416

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International Journal of Sustainable Energy Vol. 26, No. 1, March 2007, 1–17

Power optimization of a regenerated closed variable-temperature heat reservoir Brayton cycle Lingen Chen*†, Fengrui Sun† and Chih Wu‡ †Postgraduate School, Naval University of Engineering, Wuhan 430033, People’s Republic of China ‡Mechanical Engineering Department, US Naval Academy, Annapolis, MD21402, USA (Received 19 August 2004) In this paper, the power output of the cycle is taken as an objective for performance analysis and optimization of an irreversible regenerated closed Brayton cycle coupled to variable-temperature heat reservoirs in the viewpoint of finite time thermodynamics or entropy generation minimization. The analytical formulae about the relations between power output and pressure ratio are derived with the heat resistance losses in the hot- and cold-side heat exchangers and the regenerator, the irreversible compression and expansion losses in the compressor and turbine, the pressure drop losses at the heater, cooler and regenerator as well as in the piping and the effect of the finite thermal capacity rate of the heat reservoirs. The maximum power output optimization is performed in two aspects. The first is to search the optimum heat conductance distribution corresponding to the optimum power output among the hot- and cold-side of the heat exchangers and the regenerator for a fixed total heat exchanger inventory. The second is to search the optimum thermal capacitance rate matching corresponding to the optimum power output between the working fluid and the high-temperature heat source for a fixed ratio of the thermal capacitance rates of two heat reservoirs. The influences of some design parameters on the optimum heat conductance distribution, the optimum thermal capacitance rate matching and the maximum power output, which include the inlet temperature ratio of the heat reservoirs, the efficiencies of the compressor and the turbine, and the pressure recovery coefficient, are provided by numerical examples. The power plant design with optimization leads to a smaller size, including the compressor, turbine, and the hot- and cold-side heat exchangers and the regenerator. When the heat transfers between the working fluid and the heat reservoirs are carried out ideally, the pressure drop loss may be neglected, and the thermal capacity rates of the heat reservoirs become infinite. The results of this paper become those obtained in recent literature. Keywords: Closed regenerated Brayton cycle; Irreversible cycle; Finite heat capacity heat reservoirs; Power output; Performance optimization

1.

Introduction

From the time the finite time thermodynamics (FTT) or entropy generation minimization (EGM) was advanced, much work has been carried out for the performance analysis and optimization of finite time processes and finite size devices (Curzon and Ahlborn 1975, Bejan 1982, Andersen 1983, Sieniutycz and Salamon 1990, de Vos 1992, Radcenco 1994, Andresen *Corresponding author. Emails: [email protected] and [email protected]

International Journal of Sustainable Energy ISSN 1478-6451 print/ISSN 1478-646X online © 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/14786450701259416


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1996, Feidt 1996a, Bejan 1996, Haffmann et al. 1997, Berry et al. 1999, Chen et al. 1999a; Bejan 2000, Gordon and Ng 2000, Salamon et al. 2001, Denton 2002, Sieniutycz 2003, Durmayaz et al. 2004, Chen and Sun 2004, Chen 2005). The thermodynamic performance of simple (Wu 1991, Ibrahim et al. 1991, Fedit 1996b, Chen et al. 1997a, Cheng and Chen 1997, 1998a, 1998b, Radcenco et al. 1998, Cheng and Chen 1999, Hung et al. 2000, Chen et al. 2001a), regenerated (Roco et al. 1997, Chen et al. 1997b, 1999b, 2001b, 2001b, Kaushik and Tyagi 2002, Chen et al. 2004), and intercooled (Cheng and Chen 2000), endoreversible (Wu C. 1991, Ibrahim et al. 1991, Cheng and Chen 1998, 2000, Chen et al. 2001) or irreversible (Chen et al. 1997a, Fedit 1996b, Cheng and Chen. 1997, 1998b, 1999, Huang et al. 2000, Radcenco et al. 1998, Chen et al. 1997, 1999b, Roco et al. 1997, Chen et al. 2001b, 2001c, Kushik and Tyagi 2002, Chen et al. 2004), (Wu 1991, Ibrahim et al. 1991, Feidt 1996b, Cheng and Chen 1997, Roco et al. 1997, Chen et al. 1997a, 1997b, 1999b, 2001b, 2001c, Cheng and Chen 1998a, 1998b, 1999, 2000, Hung et al. 2000, Chen et al. 2001a) closed or open (Chen et al. 1997b, Roco et al. Radcenco et al. 1998, Chen et al. 1999b, 2001b, 2001c, Kaushik and Tyagi 2002, Chen et al. 2004), Brayton cycles have also been analyzed and optimized for the power, specific power, power density, efficiency and ecological optimization objectives using the heat transfer irreversibility and/or internal irreversibilities. Goktum andYavuz (1999), Erbay et al. (2001), Kaushik et al. (2003) and Tyagi et al. (2003) analyzed and optimized the performance of an irreversible regenerative closed Brayton cycle with isothermal heat addition. Among those works, Chen et al. (2003) analyzed the performance of irreversible regenerated Brayton cycle coupled with variable-temperature heat reservoirs and derived the power and efficiency expressions of the cycle. The further step of this paper (Chen et al.1999b) is to optimize the power output of an irreversible closed regenerated Brayton cycle coupled to variable-temperature heat reservoirs using the theory of thermodynamic optimization by searching the optimum heat conductance distributions among the three heat exchangers (the hot- and cold-side heat exchangers, and the regenerator) for a fixed total heat exchanger inventory, and by searching the optimum thermal capacitance rate matching between the working fluid and the high-temperature heat source for the fixed ratio of the thermal capacitance rates of two heat reservoirs. In the analysis, the heat resistance losses in the three heat exchangers, the irreversible compression and expansion losses in the compressors and the turbine, and the pressure drop loss at the heater, cooler and regenerator as well as in the piping, are taken into account. The effects of some cycle parameters on the cycle optimum performance are analyzed by using detailed numerical examples.

2.

Model cycle and analytical relation

Consider an irreversible regenerated closed Brayton cycle 1–2–3–4–1 coupled to variabletemperature heat reservoirs as shown in figure 1. The high-temperature (hot-side) heat reservoir is considered with thermal capacity rate CH and the inlet and outlet temperatures of the heating fluid are THin and THout , respectively. The low-temperature (cold-side) heat reservoir is considered with thermal capacity rate CL and the inlet and outlet temperatures of the cooling fluid are TLin and TLout , respectively. Processes 1–2 and 3–4 are non-isentropic adiabatic compression and expansion processes in the compressor and the turbine. Process 2–5 is an isobaric heat absorption process in the regenerator, 4–6 an isobaric heat rejection process in the regenerator. Process 5–3 is an isobaric heat absorption process in the hot-side heat exchanger, process 6–1 an isobaric heat rejection process in the cold-side heat exchanger. Processes 1–2s and 3–4s are isentropic adiabatic representing the processes in an ideal compressor and an


Power optimization of a regenerated closed variable

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Figure 1. T-s diagram of an irreversible regenerated Brayton cycle.

ideal turbine. The pressure drop p at the heater, cooler and regenerator as well as in the piping is reflected using pressure recovery coefficients, i.e., D1 = p3 /p2

(1)

D2 = p1 /p4

(2)

The irreversibility in the non-isentropic compression and expansion processes is reflected by using the efficiencies of the compressor and the turbine, that is, ηc = (T2s − T1 )/(T2 − T1 )

(3)

ηt = (T3 − T4 )/(T3 − T4s )

(4)

Consider the irreversible cycle 1–2–3–4. Assuming that the heat exchangers between the working fluid, the heat reservoirs and the regenerator are counter-flow, the heat conductances (heat transfer surface area and heat transfer coefficient product) of the hot- and cold-side heat exchangers and the regenerator are UH , UL and UR , and the thermal capacity rate (mass flow rate and specific heat product) of the working fluid is Cwf . According to the properties of the heat transfer process (heat reservoirs, working fluid and heat exchangers) the rate (QH ) at which heat is transferred from heat source to working fluid, the rate (QL ) at which heat is rejected from the working fluid to the heat sink and the rate (QR ) of heat regenerated in the regenerator are, respectively, given by: QH = Cwf (T3 − T5 ) = CH min f EH (THin − T5 )

(5)

QL = Cwf (T6 − T1 ) = CL min EL (T6 − TLin )

(6)

QR = Cwf (T4 − T6 ) = Cwf (T5 − T2 ) = Cwf ER (T4 − T2 )

(7)

where EH , EL and ER are, respectively, the effectiveness of the hot- and cold-side heat exchangers and the regenerator, and are defined as: EH =

1 − exp[−NH (1 − CH min /CH max )] 1 − (CH min /CH max ) exp[−NH (1 − CH min /CH max )]

(8)

EL =

1 − exp[−NL (1 − CL min /CL max )] 1 − (CL min /CL max ) exp[−NL (1 − CL min /CL max )]

(9)

ER = NR /(NR + 1),

(10)


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where CH min and CH max are, respectively, the smaller and the larger of the two capacitance rates CH and Cwf , and CL min and CL max are, respectively, the smaller and the larger of the two capacitance rates CL and Cwf , NH , NL and NR are the number of heat transfer units of the hotand cold-side heat exchangers and the regenerator, respectively, i.e., NH = UH /CH min ,

NL = UL /CL min ,

NR = UR /Cwf

(11)

CH min = min{CH , Cwf },

CH max = max{CH , Cwf }

(12)

CL min = min{CL , Cwf },

CL max = max{CL , Cwf }

(13)

Defining the working fluid isentropic temperature ratio (x) for the compressor gives: x = T2s /T1 = (p2 /p1 )m = β m ,

(14)

where β is the compressor pressure ratio and m = (k − 1)/k, k is the ratio of specific heats. According to the definition of pressure recovery coefficient equations (1) and (2), one has the working fluid isentropic temperature ratio for the turbine as follows: T3 /T4s = (p3 /p4 )m = x(D1 D2 )m = x/D,

(15)

where D = (D1 D2 )−m . The power output and the efficiency of the cycle are defined as: W = QH − Q L

(16)

η = 1 − QL /QH

(17)

After some substitutions one can yield the dimensionless power output [W = W/(Cwf TLin )] and the thermal efficiency (η) of the cycle as follows (Chen et al.1999b):

W =

Cwf {{ηc [Cwf − (1 − ηt + ηt x −1 D)(Cwf ER + CL min EL − CL min EL ER )] −(x − 1 + ηc )(Cwf − CL min EL )[ER + (1 − 2ER )(1 − ηt + ηt x −1 D)]}CH min EH τ − {(x − 1 + ηc )[CH min EH (1 − ER ) + ER Cwf + (1 − ηt + ηt x −1 D) (1 − 2ER )(Cwf − CH min EH )] − ηc [Cwf − (1 − ηt + ηt x −1 D)ER (Cwf −CH min EH )]}CL min EL } 2 CL {ηc [Cwf − Cwf ER (Cwf − CH min EH )(1 − ηt + ηt x −1 D)] − (x − 1 + ηc ) (Cwf − CL min EL )[Cwf ER + (Cwf − CH min EH )(1 − 2ER )(1 − ηt + ηt x −1 D)]}

η =1−

CL min EL {(1 − ηt + ηt x −1 D)(1 − ER )(CH min EH ηc τ − ηc [Cwf −ER (Cwf − CH min EH )(1 − ηt + ηt x −1 D)] − (x − 1 + ηc )[ER Cwf +(1 − ηt + ηt x −1 D)(1 − 2ER )(Cwf − CH min EH )]} CH min EH {{ηc Cwf [1 − ER (1 − ηt + ηt x −1 D)] − (x − 1 + ηc )Cwf −CL min EL )[ER + (1 − 2ER )(1 − ηt + ηt x −1 D)]}τ − (x − 1 + ηc ) (1 − 2ER )CL min EL }

,

(18)

(19)

where τ = THin /TLin is the cycle heat reservoir inlet temperature ratio.

3.

Optimum distribution of heat conductance

For the fixed heat conductances UH , UL and UR , Equation (18) shows that there exists an optimum working fluid temperature ratio (xopt ) (the corresponding optimum pressure ratio



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is βopt ), which leads to the maximum dimensionless power (W max ) (Chen et al. 1999b). In the practical design, UH , UL and UR are changeable. For the fixed pressure ratio (β), there exists a pair of optimum distributions among the heat conductance of hot- and cold-side heat exchangers and the regenerator for the fixed total heat exchanger inventory, which leads to the optimum dimensionless power (Wmax ). The optimum pressure ratio (βopt ) and a pair of optimum distributions lead to the maximum optimum (double-maximum) dimensionless power (W max,max ). They may be determined using numerical calculation. For the fixed hot- and cold-side heat exchanger inventory (UT ), i.e. for the constraint UH + UL + UR = UT ,

(20)

defining the hot-side heat conductance distribution (uH ) and the cold-side heat conductance distribution (uL ) as

Figure 2.

Figure 3.

uH = UH /UT

(21)

uL = UL /UT

(22)

Dimensionless power versus heat conductance distributions.

Cycle thermal efficiency versus heat conductance distributions.


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leads to UH = uH UT

(23)

UL = uL UT

(24)

UR = (1 − uL − uH )UT

(25)

The power optimization is performed using numerical calculation. In the calculation, k = 1.4, CH = CL = 1.2kW/K and Cwf = 1.3kW/K are set. The dimensionless power (W ) and the thermal efficiency (η) of the cycle versus the hotside heat conductance distribution (uH ) and the cold-side heat conductance distribution (uL ) with β = 6.0, UT = 5.0kW/K, τ = 4.0, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 are shown in figures 2 and 3. The two three-dimensional diagrams show the existence of the maximum dimensionless power output and the maximum efficiency. One can see from figures 2 and 3 that for a fixed β, there exists a pair of uHopt and uLopt , which leads to the optimum dimensionless power (W max ), and there exists another pair of uHopt and uLopt , which leads to the optimum efficiency (ηmax ). The vertical plane in the two diagrams is the one with uH + uL = 1. If uH + uL = 1, the regenerated Brayton cycle becomes a simple one. If uH + uL is larger than one, the practical plant does not exist. Therefore, in the numerical calculation, the following

Figure 4.

Figure 5. ratio.

Optimum dimensionless power versus cycle heat reservoir temperature ratio and pressure ratio.

Optimum hot-side heat conductance distribution versus cycle heat reservoir temperature ratio and pressure



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Figure 6. Optimum cold-side heat conductance distribution versus cycle heat reservoir temperature ratio and pressure ratio.

Figure 7.

Figure 8.

Optimum dimensionless power versus heat exchanger inventory and pressure ratio.

Optimum hot-side heat conductance distribution versus heat exchanger inventory and pressure ratio.


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Figure 9.

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Optimum cold-side heat conductance distribution versus heat exchanger inventory and pressure ratio.

conditions should be satisfied: uH ≤ 1

(26)

uL ≤ 1

(27)

uH + uL ≤ 1.

(28)

The influence of cycle heat reservoir temperature ratio (τ ) on the optimum dimensionless power (W max ) versus pressure ratio (β) with UT = 5.0kW/K, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 4. The corresponding optimum heat conductance distributions (uHopt and uLopt ) are shown in figures 5 and 6. The influence of the total heat exchanger inventory (UT ) on the optimum dimensionless power (W max ) versus pressure ratio (β) with τ = 5.0, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 7. The corresponding optimum heat conductance distributions (uHopt and uLopt ) are shown in figures 8 and 9. The influence of the efficiency of compressor (ηc ) on the optimum power (W max ) versus cycle pressure ratio (β) with τ = 5.0, UT = 5.0kW/K, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 10. The corresponding optimum heat conductance distributions (uHopt and uLopt ) are shown in figures 11 and 12.

Figure 10.

Optimum dimensionless power versus compressor efficiency and pressure ratio.



Figure 11.

Optimum hot-side heat conductance distribution versus compressor efficiency and pressure ratio.

Figure 12.

Optimum cold-side heat conductance distribution versus compressor efficiency and pressure ratio.

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The influence of the efficiency of turbine (ηt ) on the optimum power (W max ) versus cycle pressure ratio (β) with τ = 5.0, UT = 5.0kW/K, ηc = 0.8 and D1 = D2 = 0.96 is shown in figure 13. The corresponding optimum heat conductance distributions (uHopt and uLopt ) are shown in figures 14 and 15. The influence of the pressure recovery coefficients (D1 = D2 ) on the optimum dimensionless power (W max ) versus β with τ = 5.0, UT = 5.0kW/K, ηc = 0.8, and ηt = 0.9 is shown in figure 16. The corresponding optimum heat conductance distributions (uHopt and uLopt ) are shown in figures 17 and 18. The parts of dotted line in Figures 4–18 have no meaning in practice because they correspond to the condition of uH + uL ≥ 1. They are used only to observe the numerical potentials. The numerical results show that there exists a pair of optimum heat conductance distributions and an optimum pressure ratio corresponding to double maximum power output. For different pressure ratio, there exist different optimum heat conductance distributions corresponding to optimum power. In general, the optimum pressure ratio approaches the critical condition uH + uL = 1. Hence, the practical choice of the cycle pressure ratio should be less than the optimum value for a regenerated Brayton cycle.


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Figure 13.

Optimum dimensionless power versus turbine efficiency and pressure ratio.

Figure 14.

Optimum hot-side heat conductance distribution versus turbine efficiency and pressure ratio.

Figure 15.

Optimum cold-side heat conductance distribution versus turbine efficiency and pressure ratio.

The optimum dimensionless power output (W max ) increases with increases in the cycle heat reservoir temperature ratio (τ ), the total heat exchanger inventory (UT ), the efficiencies of compressor and turbine (ηc and ηt ) and the pressure recovery coefficients (D1 = D2 ).



Figure 16.

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Optimum dimensionless power versus pressure recovery coefficients and pressure ratio.

Figure 17.

Optimum hot-side heat conductance distribution versus pressure recovery coefficients and pressure ratio.

Figure 18. ratio.

Optimum cold-side heat conductance distribution versus pressure recovery coefficients and pressure


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The optimum hot-side heat conductance distribution (uHopt ) increases with a decrease in the cycle heat reservoir temperature ratio (τ ) and increases in the total heat exchanger inventory (UT ), the efficiencies of compressor and turbine (ηc and ηt ) and the pressure recovery coefficients (D1 = D2 ). The optimum cold-side heat conductance distribution (uLopt ) increases with decreases in the cycle heat reservoir temperature ratio (τ ), the total heat exchanger inventory (UT ), the efficiencies of compressor and turbine (ηc and ηt ) and the pressure recovery coefficients (D1 = D2 ). Both the optimum hot-side heat conductance distribution (uHopt ) and the optimum cold-side heat conductance distribution (uLopt ) are increasing functions of the cycle pressure ratio (β) in general.

4.

Optimum matching between the thermal capacity rates of the working fluid and reservoirs

For the fixed CH /CL , there exists an optimum matching of thermal capacitance rates between working fluid and high-temperature heat source, i.e. there exists an optimum value of Cwf /CL , which leads to the optimum dimensionless power output (P max ). They may be determined using numerical calculation. In the numerical calculation, uH = 0.3, uL = 0.5, CH = 0.8kW/K and k = 1.4 are set. The influence of the ratio (CH /CL ) of thermal capacity rates of the high- and low- temperature heat reservoirs on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with UT = 5kW/K, β = 8, τ = 5.0, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 19. The influence of heat exchanger inventory (UT ) on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with CH /CL = 1.2, β = 8, τ = 5.0, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 20. The influence of cycle pressure ratio (β) on the optimum matching between the thermal capacitance rates of the working fluid and the heat reservoirs with UT = 5kW/K, CH /CL = 1.2, τ = 5.0, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 21.

Figure 19. Dimensionless power versus thermal capacity rate ratio of heat reservoirs and thermal capacity rate matching of the working fluid and low-temperature reservoir.



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Figure 20. Dimensionless power versus heat exchanger inventory and thermal capacity rate matching of the working fluid and low-temperature reservoir.

Figure 21. Dimensionless power versus compressor pressure ratio and thermal capacity rate matching of the working fluid and low-temperature reservoir.

The influence of cycle heat reservoir inlet temperature ratio (τ ) on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with UT = 5kW/K, β = 8, CH /CL = 1.2, ηc = 0.8, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 22. The influence of the efficiency of the compressor (ηc ) on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with UT = 5kW/K, β = 8, CH /CL = 1.2, τ = 5.0, ηt = 0.9 and D1 = D2 = 0.96 is shown in figure 23. The influence of the efficiency of the compressor (ηt ) on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with UT = 5kW/K, β = 8, CH /CL = 1.2, τ = 5.0, ηc = 0.8 and D1 = D2 = 0.96 is shown in figure 24. The influence of the pressure recovery coefficient (D1 = D2 ) on the optimum matching between the thermal capacity rates of the working fluid and the heat reservoirs with UT = 5kW/K, β = 8, CH /CL = 1.2, τ = 4.0, ηc = 0.8 and ηt = 0.9 is shown in figure 25. The optimum matching between the thermal capacitance rates of the working fluid and the heat reservoirs (Cwf /CL )opt increases with increases in UT , τ , CH /CL , ηc , ηt and D1 = D2 and a decrease in β. The analytical results about dimensionless power output (W ) with EH = EL = 1 of this paper replicate the results given in Refs. (Woods 1991, Woods et al. 1991, Frost et al. 1992,


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Figure 22. Dimensionless power versus heat reservoir inlet temperature ratio and thermal capacity rate matching of the working fluid and low-temperature reservoir.

Figure 23. Dimensionless power versus compressor efficiency and thermal capacity rate matching of the working fluid and low-temperature reservoir.

Figure 24. Dimensionless power versus turbine efficiency and thermal capacity rate matching of the working fluid and low-temperature reservoir.



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Figure 25. Dimensionless power versus pressure recovery coefficients and thermal capacity rate matching of the working fluid and low-temperature reservoir.

Horlock and Woods, 2000) by using a conventional method in which irreversible Brayton cycles without external heat transfer irreversibility were examined.

5.

Conclusion

The performance of an irreversible regenerated Brayton cycle coupled to variable-temperature heat reservoirs with heat transfer irreversibility in the hot- and cold-side heat exchangers and the regenerator, irreversible compression and expansion losses in the compressor and turbine, the pressure drop loss in the piping and the effect of the finite thermal capacity rate of the heat reservoirs was optimized by taking the power output as the optimization objective. The optimization is performed by optimizing the distribution among the heat conductances of the hot- and cold-side heat exchangers and the regenerator for the fixed total heat exchanger inventory, and by optimizing the matching between the thermal capacitance rates of the working fluid and the heat reservoirs. The influences of various parameters on the optimum power, optimum heat conductance ratios, and optimum thermal capacitance rate ratio are analyzed. The optimum distribution of heat conductance will lead to the minimum heat exchanger inventory for the fixed power output. Therefore, the optimization carried out herein will lead to a regenerated gas turbine power plant design with smaller size and higher efficiency. The analysis and optimization may provide guidelines for optimal design in terms of power, thermal efficiency and engine size for real closed regenerated gas turbine power plants. Acknowledgements This paper is supported by Program for New Century Excellent Talents in University of P. R. China (Project No. NCET-04-1006) and The Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project No. 200136). References Andresen B., Finite-Time Thermodynamics, Physics Laboratory II, University of Copenhagen, 1983. Andresen, B., Finite-time thermodynamics and thermodynamic length. Rev. Gen. Therm, 1996, 35(418/419), 647–650. Bejan, A., Entropy Generation through Heat and Fluid Flow, 1982 (Wiley, New York).


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