Performance comparison of an endoreversible closed variable temperature heat reservoir Brayton cycle under maximum power density and maximum power conditions

Energy Conversion and Management 43 (2002) 33±43

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Performance comparison of an endoreversible closed variable temperature heat reservoir Brayton cycle under maximum power density and maximum power conditions Lingen Chen a,*, Junlin Zheng a, Fengrui Sun a, Chih Wu b a b

Faculty 306, Naval University of Engineering, Wuhan 430033, People's Republic of China Department of Mechanical Engineering, US Naval Academy, Annapolis, MD 21402, USA Received 5 September 2000; accepted 18 December 2000

Abstract In this paper, the power density, de®ned as the ratio of power output to maximum speci®c volume in the cycle, is taken as the objective for performance analysis of an endoreversible closed Brayton cycle coupled to variable temperature heat reservoirs in the viewpoint of ®nite time thermodynamics or entropy generation minimization. The analytical formulae about the relations between power density and pressure ratio are derived with heat resistance losses in the hot and cold side heat exchangers. The obtained results are compared with those results obtained by using the maximum power criterion. The in¯uences of some design parameters on the maximum power density are provided by numerical examples, and the advantages and disadvantages of maximum power density design are analyzed. The power plant design with maximum power density leads to a higher eciency and smaller size. When the heat transfer is e€ected ideally and the thermal capacity rates of the two heat reservoirs are in®nite, the results of this paper become those obtained in recent literature. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Closed Brayton cycle; Endoreversible cycle; Power density; Power; Performance comparison

1. Introduction Since ®nite time thermodynamics (FTT) or entropy generation minimization was advanced [1± 3], much work has been performed for the performance analysis and optimization of ®nite time processes and ®nite size devices [4±10]. The FTT performance of the Brayton cycle has been also *

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

0196-8904/02/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 0 0 3 - 6

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L. Chen et al. / Energy Conversion and Management 43 (2002) 33±43

analyzed with the power, speci®c power, eciency and ecological optimization objectives with heat transfer irreversibility and/or internal irreversibilities [11±28]. Sahin et al. [29] taking the power density, de®ned as the ratio of power output to the maximum speci®c volume in the cycle, as optimization objective, analyzed the optimum performance of an ideal reversible simple Brayton cycle free of heat transfer and any other irreversibilities. Yavuz [30] investigated the maximum power density performance for internally irreversible Brayton cycle free of heat transfer irreversibility. Medina et al. [31] applied the maximum power density method to an internally irreversible regenerative Brayton cycle free of heat transfer irreversibility. Sahin et al. [32] applied the maximum power density method to an internally irreversible regenerative reheating Brayton cycle free of heat transfer irreversibility. The power density objective can be used to optimize the cycle performance including engine size e€ects and was applied to an ideal reversible Ericsson cycle [33] and an Atkinson cycle [34] free of any irreversibility. The power density objective was also applied to an endoreversible Carnot heat engine [35] with only the external heat transfer irreversibility, irreversible Carnot heat engine [36] and a combined cycle Carnot cycle [37] with internal irreversibility and external heat transfer irreversibility. The further step of this paper is to analyze the endoreversible Brayton cycle performance using the power density objective with considerations of the heat transfer irreversibility in the hot and cold side heat exchangers and the e€ects of ®nite thermal capacitance rates. The obtained results are compared with those results obtained by using the maximum power criterion. If the heat transfer in the hot and cold side heat exchangers is performed ideally and thermal capacity rates of the two heat reservoirs are in®nite, the results of this paper become those results obtained by Sahin et al. [29].

2. Analytical relation Consider an endoreversible closed Brayton cycle coupled to variable temperature heat reservoirs, as shown in Fig. 1. Processes 1±2 and 3±4 are adiabatic, and processes 2±3 and 4±1 are isobaric. Assuming that the heat exchangers are counter ¯ow, the heat conductance (heat transfer surface area and heat transfer coecient product) of the hot and cold side heat exchangers are UH and UL , and the thermal capacity rate (mass ¯ow rate and speci®c heat product) of the working ¯uid is Cwf . The high temperature (hot side) heat reservoir is considered with thermal capacity rate CH , and the inlet and outlet temperatures of the heating ¯uid 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 ¯uid are TLin and TLout , respectively. Processes 1±2 and 3±4 are adiabatic and are assumed isentropic, that is, the eciencies of the compressor and the turbine are unity, gc ˆ gt ˆ 1. Therefore, according to the properties of the heat transfer processes, heat reservoirs, working ¯uid and heat exchangers, the rate …QH † at which heat is transferred from the heat source to the working ¯uid, and the rate …QL † at which heat is rejected from the working ¯uid to the heat sink are, respectively, given by: QH ˆ Cwf …T3

T2 † ˆ CH min EH …THin

QL ˆ Cwf …T4

T1 † ˆ CL min EL …T4

T2 † TLin †

…1† …2†

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35

Fig. 1. T±s diagram of the endoreversible Brayton cycle.

where EH and EL are, respectively, the e€ectiveness of the hot and cold side heat exchangers, de®ned as: EH ˆ EL ˆ

1

1

exp‰ NH1 …1 CH min =CH max †Š …CH min =CH max † exp‰ NH1 …1 CH min =CH max †Š 1

1

exp‰ NL1 …1 CLmin =CLmax †Š …CLmin =CLmax † exp‰ NL1 …1 CLmin =CLmax †Š

…3† …4†

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 . The numbers of heat transfer units, NH1 and NL1 , are based on the minimum thermal capacitance rates, that is: NH1 ˆ UH =CH min ;

NL1 ˆ UL =CL min

…5†

CH min ˆ minfCH ; Cwf g;

CH max ˆ maxfCH ; Cwf g

…6†

CL min ˆ minfCL ; Cwf g;

CL max ˆ maxfCL ; Cwf g

…7†

Applying the second law of thermodynamics to the endoreversible cycle 1±2±3±4 gives T1 T3 ˆ T2 T4 . De®ning the working ¯uid temperature ratio …x† for the compressor gives: x ˆ T2 =T1 ˆ T3 =T4 ˆ …p2 =p1 †m ˆ pm

…8†

where p is the compressor pressure ratio and m ˆ …k 1†=k, where k is the ratio of speci®c heats. The power output and the eciency of the cycle are de®ned as: W ˆ QH gˆ1

QL QL =QH

…9† …10†

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Combining Eqs. (1)±(10) gives the outlet temperatures T2 and T4 of the compressor and the turbine, the dimensionless power output …W ˆ W =…Cwf TLin †† and the thermal eciency …g† of the cycle as follows : T2 ˆ

CH min EH …Cwf CL min EL † THin ‡ xCwf CL min EL TLin 2 Cwf …Cwf CH min EH †…Cwf CL min EL†

…11†

T4 ˆ

x 1 CH min EH Cwf THin ‡ CL min EL …Cwf CH min EH † TLin 2 Cwf …Cwf CH min EH †…Cwf CL min EL†

…12†

W ˆ

CH min CL min EH EL ‰…1 x 1 †s x ‡ 1Š Cwf …CH min EH ‡ CL min EL † CH min CL min EH EL

…13†

gˆ1

x

1

…14†

Additionally, m4 =m1 ˆ T4 =T1 ˆ …T4 =T2 †…T2 =T1 † ˆ x…T4 =T2 †

…15†

Substituting Eqs. (11) and (12) into Eq. (15) yields: m4 CH min EH Cwf THin ‡ xCL min EL …Cwf ˆ m1 xCwf CL min EL TLin ‡ CH min EH …Cwf

CH min EH † TLin CL min EL † THin

…16†

The power density (P) is de®ned as …17†

P ˆ W =m4

Substituting Eqs. (13) and (16) into Eq. (17) yields the dimensionless power density ‰P ˆ P = …Cwf TLin =m1 †Š as follows: Pˆ

CH min CL min EH EL ‰xCwf CL min EL ‡ CH min EH …Cwf CL min EL †sŠ‰…1 x 1 †s ‰Cwf …CH min EH ‡ CL min EL † CH min CL min EH EL Š‰CH min EH Cwf s ‡ xCL min EL …Cwf

x ‡ 1Š CH min EH †Š …18†

where s ˆ THin =TLin is the cycle heat reservoir inlet temperature ratio. If the heat transfers between the working ¯uid and the heat reservoirs can be performed ideally, i.e. UH ˆ UL ! 1; EH ˆ EL ˆ 1:0, Eq. (18) becomes: Pˆ

CH min CL min ‰xCwf CL min ‡ CH min …Cwf CL min †sŠ‰…1 x 1 †s ‰Cwf …CH min ‡ CL min † CH min CL min Š‰CH min Cwf s ‡ xCL min …Cwf

x ‡ 1Š CH min †Š

…19†

If the heat transfers between the working ¯uid and the heat reservoirs can be performed ideally and the thermal capacity rates of the two heat reservoirs are in®nite, Eq. (18) becomes:    x 1 P ˆx 1 1 …20† s x The maximum dimensionless power density …P max † and the corresponding optimum working ¯uid temperature ratio …xopt † and the eciency …gP † are as follows:

L. Chen et al. / Energy Conversion and Management 43 (2002) 33±43

P max ˆ …s

1†2 =…4s†

37

…21†

xopt ˆ …1 ‡ s†=2

…22†

gP ˆ …s

…23†

1†=…s ‡ 1†

Eqs. (21)±(23) are the major results obtained by Sahin et al. [29] in which an ideal heat transfer Brayton cycle was examined. In these equations, s ˆ THin =TLin ˆ T3 =T1 . Eqs. (19) and (20) indicate that ®nite thermal capacity rates of the two heat reservoirs do a€ect the performance of the cycle. If and only if Cwf 6 CH and Cwf 6 CL ; CH min ˆ CL min ˆ Cwf , and the performance is independent of the thermal capacity rates of the two heat reservoirs.

3. Performance comparison for power and power density objectives To see the advantages and disadvantages of maximum power density design, detailed numerical examples are provided and are compared with those for the maximum power objective. In the calculation, Cwf ˆ 1:5 kW/K, CH ˆ CL ˆ 1 kW/K and k ˆ 1:4 are set. For a varying cycle thermal eciency …g†, the normalized dimensionless power …W =W max † and the normalized dimensionless power density …P =P max † with EH ˆ EL ˆ 0:9 and s ˆ 4:0 are plotted in Fig. 2. As one can see, the eciency …gP † at maximum power density is greater than the eciency …gW † at maximum power.

Fig. 2. Normalized power density and normalized power versus thermal eciency.

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Fig. 3. Eciency versus heat reservoir inlet temperature ratio.

Fig. 4. Normalized speci®c volume versus heat reservoir inlet temperature ratio.

Variations of the two eciencies gW and gP with s are illustrated in Fig. 3 with EH ˆ EL ˆ 0:9. Variations in the dimensionless maximum speci®c volume …m4 =m1 †P at the maximum power density and …m4 =m1 †W at the maximum power in the cycle with s are shown in Fig. 4 with EH ˆ EL ˆ 0:9. The variation in normalized maximum speci®c volume di€erence …m4 m2 †P =…m4 m2 †W in the cycle with s is shown in Fig. 5 with EH ˆ EL ˆ 0:9. The variation in normalized pressure ratio …p2 =p1 †P =…p2 =p1 †W in the cycle with s is shown in Fig. 6 with EH ˆ EL ˆ 0:9. Variations of the two eciencies gW and gP with EH ˆ EL are shown in Fig. 7 with s ˆ 4:0. The variations in the dimensionless maximum speci®c volume …m4 =m1 †P and …m4 =m1 †W in the cycle with EH ˆ EL are shown in Fig. 8 with s ˆ 4:0. The variation in normalized maximum speci®c volume di€erence …m4 m2 †P =…m4 m2 †W in the cycle with EH ˆ EL is shown in Fig. 9. The variation in

L. Chen et al. / Energy Conversion and Management 43 (2002) 33±43

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Fig. 5. Normalized maximum speci®c volume di€erence versus inlet heat reservoir temperature ratio.

Fig. 6. Normalized pressure ratio versus heat reservoir inlet temperature ratio.

normalized pressure ratio …p2 =p1 †P =…p2 =p1 †W in the cycle with EH ˆ EL is shown in Fig. 10 with s ˆ 4:0. Figs. 3 and 7 show that the eciency …gP † at maximum power density is always larger than that …gW † at maximum power, gP increases with increases of s, EH and EL , and gW increases with increases of s but it is independent of EH and EL . Figs. 4 and 8 show that an engine working at maximum power density is smaller than one working at maximum power because its maximum speci®c volume is always smaller. The maximum speci®c volumes for the two cases increase with the increases of EH , EL and s, but the maximum speci®c volume at maximum power density increases more slowly than that at maximum power as s, EH and EL increase.

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Fig. 7. Eciency versus e€ectiveness of heat exchangers.

Fig. 8. Normalized speci®c volume versus e€ectiveness of heat exchangers.

Figs. 5 and 9 show that the ratio of maximum speci®c volume di€erence at maximum power density to that at maximum power is less than one in the reasonable ranges of s, EH and EL . That is, an engine design using parameters at maximum power density has the advantage of smaller size and higher eciency. Figs. 6 and 10 show that the maximum power density design requires a higher pressure ratio than the maximum power design.

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Fig. 9. Normalized maximum speci®c volume di€erence versus e€ectiveness of heat exchangers.

Fig. 10. Normalized pressure ratio versus e€ectiveness of heat exchangers.

4. Conclusion The performance of an endoreversible Brayton cycle coupled to variable temperature heat reservoirs with heat transfer irreversibility in the hot and cold side heat exchangers was analyzed by taking the power density as the optimization objective. Comparisons between maximum power density performance and maximum power performance were made. The maximum power density design has the advantages of smaller size and higher eciency, but it requires a higher pressure ratio than the maximum power design. All of the analytical results with EH ˆ EL ˆ 1 and CH ˆ CL ! 1 of this paper replicate the results given by Sahin et al. [29] in which an ideal Brayton

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