Power optimization of an irreversible regenerative Brayton cycle with isothermal heat addition
Journal of Thermal Engineering 2015, Vol. 1, Issue 4, pp. 279-286; doi.org/10.18186/jte.44164
Abstract
Keywords: Thermodynamic Optimization; Irreversible Brayton cycle; Regenerator; Power and Efficiency
1. Introduction
The most important criterion in the design of real gas power plant is not only efficiency but power output also. Leff [2] analyzed an endoreversible Brayton heat engine following Curzon and Ahlborn [1] and observed the change in Brayton cycle temperatures while altering maximum work in the cycle. Wu and Kiang [3] optimized power output of a Brayton cycle using finite time thermodynamics. Wu [4] optimized the power of an endoreversible Brayton gas heat engine. Wu & Kiang [5] integrated real compression and expansion in Brayton heat engine and found that engine power and engine efficiency are strong functions of the compressor and turbine efficiencies. Ibrahim et al. [6] performed power optimization for a closed ideal Brayton cycle in context with various boundary configurations. Cheng et al. [7] performed power optimization of an endoreversible regenerative Brayton cycle and observed decrease in maximum power and corresponding efficiency with the application of regenerator. Wu et al. [8] investigated performance of a regenerative Brayton heat engine and found that maximum non-dimensional power output of cycle increases from 0.2 to 0.4 by altering regenerator effectiveness between 0.6 and 1. Chen et al. [9] further assessed performance of regenerative Brayton cycle and found that regenerative effectiveness is the deciding factor while calculating power output of the cycle. Many other researchers [10-13] assessed
Research Article – JTEN – 2014 – 57 regenerative Brayton cycle based on endoreversible medium at compressor and compressed up to state 2. and irreversible configuration with the application of Then the working medium enters the regenerator isothermal heat additions in the view of finite time where its partial heating up to state 2R is done by the thermodynamic approach. Wang et al. [14] applied the turbine exhaust. The working medium next enters the hypothesis of finite time thermodynamics to analyze hot side heat exchanger with a pressure drop which is an irreversible closed intercooled regenerated Brayton reflected using pressure recovery coefficient, α1 = cycle and optimized the intercooler pressure ratio for p3/p2 and heated up to state 3. First heat addition (QH) optimum power and corresponding efficiency. takes place at constant pressure in hot side heat Kaushik et al. [15] performed a thermodynamic exchanger while the heat source temperature decreases analysis of an irreversible regenerative Brayton cycle from TH1 to TH2. Again, second heat addition (QH1) with isothermal heat addition and optimized the power takes place at constant temperature during process 3-4 output in context with working medium temperature. and heat source temperature decreases from TH3 to TH4.The working medium now enters the turbine and They observed an improvement of 15% in the thermal efficiency of Brayton cycle with heat addition at expands up to state 5. After expansion, the working constant temperature. Chen et al. [16] analyzed power medium enters the regenerator to transfer heat partly and efficiency of an endoreversible closed intercooled and then enters the cold side heat exchanger with a pressure drop which is reflected using another pressure regenerated Brayton cycle in the view of finite time recovery coefficient, α2 = p1/p5. The working medium thermodynamics. Wang et al. [17-19] performed is cooled up to state 1, while the heat sink temperature power optimization by altering effectiveness of increases from TL1 to TL2. Therefore, we consider the various heat exchangers for intercooled and closed Brayton cycle model 1-2-2R-3-4-5-5R-1 with regenerated Brayton cycles coupled to fixed [17] and real compression / expansion processes and pressure finite temperature [18-19] heat reservoirs based on drop irreversibilities for finite heat capacity of external endoreversible [17,18] and irreversible [19] mode.. reservoirs. Process (1-2s) and process (4-5s) are Jubeh [20] performed exergy analysis of a regenerative isentropic in nature as shown by dotted lines in Figure Brayton cycle and found appreciable increase in 1. second law efficiency at lesser pressure ratio, small environment temperature and elevated entrance temperature of expander with the introduction of two heat additions. Further, Wang et al. [21] investigated power and power density of externally irreversible Brayton cycle with two heat additions in the view of finite time thermodynamics and found the range of isothermal heat addition on various performance parameters of endoreversible Brayton cycle. On the basis of recent literature, a model of an irreversible regenerative Brayton cycle with pressure drop as supplementary irreversibility is considered in this paper and expressions for maximum power output and corresponding thermal efficiency of an irreversible regenerative Brayton cycle are obtained. The effect of effectiveness of various heat exchangers, efficiency of Fig. 1 T-S diagram for irreversible regenerative turbine and compressor, heat capacitance rates, Brayton heat cycle with isothermal heat addition isothermal pressure drop ratio and pressure recovery coefficients have been studied in detail and the results The various heat transfer rates are calculated are presented on graphs. The model analyzed in this as: paper gives lower values of power output and QH = U H AH ( LMTD) H = CH (TH 1 − TH 2 ) (1) corresponding thermal efficiency as expected. QH 1 = U H 1 AH 1 ( LMTD) H 1 = CH 1 (TH 3 − TH 4 ) (2)
2. Thermodynamic Analysis
QL = U L AL ( LMTD) L = CL (TL 2 − TL1 ) (3) An irreversible regenerative Brayton cycle QR = U R AR ( LMTD) R = CW (T2 R − T2 ) (4) model coupled with a heat source and heat sink of finite heat capacity is shown on T-S diagram in Fig. 1. In this model, state 1 is the entry point of working where,
(TH 1 − T3 ) − (TH 2 − T2 R ) ln {(TH 1 − T3 ) (TH 2 − T2 R )}
(T5 R − TL 2 ) − (T1 − TL1 ) ln {(T5 R − TL 2 ) (T1 − TL1 )}
(T5 − T2 R ) − (T5 R − T2 ) ( LMTD) R = ln {(T5 − T2 R ) (T5 R − T2 )}
where εH, εH1, εL and εR are the effectiveness of the isobaric heat source side, isothermal heat source side, sink side and regenerative side heat exchangers respectively and can be presented as: − N (1−C
1 − e H H ,min H ,max C − N (1−C C ) 1 − H ,min e H H ,min H ,max CH ,max
1− e C − N (1−C C ) 1 − L,min e L L ,min L ,max CL,max NR 1+ NR
isothermal pressure drop ratio. Putting the values of T1, T3, T2s and T5s from equations (21) - (24) into equation (25), we obtain the quadratic equation in T2 as:
Parameters X, Y and Z are listed in Appendix-I. Solution of quadratic equation (26) is written as:
The various heat transfer rates and number of transfer units are calculated as:
Putting the value of various temperatures into equation (28), we get:
effectiveness of various heat exchangers is required. However, in general, the variations of various performance parameters with respect to effectiveness are not linear and the relation εL> εH1> εH is observed for better execution of the model.
Parameters x7, x8, y7, y8, z6 and z7 are listed in Appendix-I. Thus, optimizing equation (29) with respect
∂P to T5 i.e. = 0 and solution of this equation as: ∂T5 X T + Y1T5 + Z1 = 0
Parameters X1, Y1 and Z1 are recorded in Appendix-I. Solving equation (31) for T5, we get the optimum value of T5 as
Fig. 2 (a) Variations of Power Output with respect to effectiveness of heat exchangers
3. Results And Discussion
In order to have mathematical approval of outcome, the effects of various performance parameters viz. efficiency of turbine and compressor, effectiveness of various heat exchangers, isothermal pressure drop ratio, pressure drop recovery coefficients and heat capacitance rate of the working fluid on an irreversible regenerative Brayton heat engine model are investigated. Each one of above mentioned parameter is examined by keeping rest parameters constant as εH= εH1=εL= εR=0.75, TH1=1000, TH3=1250 K, TL1=300 K, ηt= ηc=0.8, CW=1.05 kWK-1, CH= CH1 =CL=1 kWK-1, UH= UH1=UL=UR=2.0 kWK-1m-2, χt=0.8, α1= α2=0.95. The obtained results are presented on graphs and discussed in detail as follows:
Fig 2(b) Variations of Thermal Efficiency with respect to effectiveness of heat exchangers
3.2. Effect of heat capacitance rates (CH, CH1, CL
and CW) The variations of various heat capacitance rates on maximum power output and corresponding thermal efficiency are shown in figures 3(a) to 3(b). It is clearly observed from these results that maximum power output and thermal efficiency increases with increase in heat capacitance rates of constant temperature source side and sink side reservoirs whereas all the performance parameters shows steep fall with the increase in heat capacitance rate of constant pressure heat source reservoir and cycle working fluid. It is also found that sink side heat capacitance rate is more dominant than constant temperature source side on all the performance parameters of the cycle. In general, the variations of various performance parameters with respect to heat capacitance rates are not linear and the relation CL>
3.1. Effect of εH, εH1, εL and εR
The variations of various effectiveness on power output and corresponding thermal efficiency are shown in figures 2(a) to 2(b). It is clearly seen from these results that maximum power output and corresponding thermal efficiency increases as the effectiveness on isothermal heat source side (εH1), heat sink side (εL) is increased while all the performance parameters decreases as isobaric heat source side effectiveness (εH) is increased. It is also found that the power output remains constant while the corresponding thermal efficiency increases as regenerator side effectiveness is increased. The results obtained can also be correlated with heat transfer area. It is required to increase the heat transfer area as the effectiveness is increased which results in increase of cost of the system. So, judicious selection of
Research Article – JTEN – 2014 – 57 CH1> CW is observed for better execution of the model. Moreover, the heat capacitance rate of constant pressure heat reservoir (CH) should be as small as possible.
3.4. Effects of isothermal pressure drop ratio
Figure 5 shows the effect of isothermal pressure ratio on various performance parameters of an irreversible regenerative Brayton heat engine cycle. It is seen from these figures that maximum power output and corresponding thermal efficiency increases as isothermal pressure drop ratio is increased. It is also seen from these figures that various performance parameters attains their maximum value at isothermal pressure drop ratio of unity which cannot be achieved in realistic Brayton heat engine cycle. Further, maximum power output and thermal efficiency reflect linear variations with isothermal pressure drop ratio.
Fig. 3 (a) Variations of Power Output with respect to heat capacitance rates
Fig. 4(a) Variations of Power Output with respect to component efficiency (ηc and ηt )
Fig. 3 (b) Variations of Thermal Efficiency with respect to heat capacitance rates
3.3. Effects of turbine and compressor efficiencies
(ηt and ηc) The variations of turbine and compressor efficiencies on power output and corresponding thermal efficiency of an irreversible regenerative Brayton heat engine cycle with finite capacity heat reservoir are shown in figures 4(a) to 4(b). It is seen from these figures that maximum power output and thermal efficiency increases with the increase in component efficiencies (ηt and ηc) which indicates that larger the component efficiency is, better the performance of the cycle. It is also found that turbine efficiency (ηt) adds more effect on the thermodynamic performance of an irreversible regenerative Brayton heat engine cycle than the compressor efficiency (ηc). Consequently, for real Brayton heat engine cycle, lots of research and investigation is still required on compressor efficiency.
Fig. 4(b) Variations of Thermal Efficiency with respect to component efficiency (ηc and ηt)
3.5. Effects of pressure recovery coefficients (α1=α2)
Fig. 6 shows the effect of pressure recovery coefficients on various performance parameters of an irreversible regenerative Brayton heat engine cycle. It is seen from these figures that maximum power output and thermal efficiency increases as the pressure drop is decreased. It is also seen from these figures that various performance parameters attains their
maximum value at zero pressure drop which cannot be achieved in realistic Brayton heat engine cycle. The power output and thermal efficiency is very low at lower values of pressure recovery coefficient (α1<0.9, α2<0.9). Hence, efforts should be made to reduce pressure drop while designing real gas power plants.
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Kumar, R.; Kaushik, S.C.; Kumar, R. Power optimization of an irreversible regenerative Brayton cycle with isothermal heat addition. Journal of Thermal Engineering 2015, Vol. 1, pp. 279-286. https://doi.org/10.18186/jte.44164