OPTIMIZATION OF A BRAYTON CYCLE AT MAXIMUM POWER OPERATING CONDITION AND AT MAXIMUM THERMAL EFFICIENCY OPERATING CONDITION

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SSN 76-580 nd nternational ongress of Mechanical Engineering Novemer -7, 0, Rieirão Preto, SP, Brazil opyright 0 y ABM OPMZAON OF A BRAYON YE A MAXMUM POER OPERANG ONDON AND A MAXMUM ERMA EFFENY

1 SSN nd nternational ongress of Mechanical Engineering Novemer -7, 0, Rieirão Preto, SP, Brazil opyright 0 y ABM OPMZAON OF A BRAYON YE A MAXMUM POER OPERANG ONDON AND A MAXMUM ERMA EFFENY OPERANG ONDON Vitor Pereira Repinaldo Unesp/Bauru Mechanical Engineering Department vitor_repinaldo@hotmail.com Santiago del Rio Oliveira Unesp/Bauru Mechanical Engineering Department santiago@fe.unesp.r Vicente uiz Scalon Unesp/Bauru Mechanical Engineering Department scalon@fe.unesp.r Astract. n this work is carried out an optimization process of a Brayton power cycle for two different conditions. he first condition corresponds to the operation at full power cycle and the second condition corresponds to the operation cycle with imum thermal efficiency. Both conditions are formulated according to the finite time thermodynamics so as to etter approximate the cycle analysis of a real process. his approximation is made considering the following irreversiilities effects: thermal resistances arising from the temperature difference etween the cycle and the two thermal reservoirs, heat leakage, in which part of the energy supplied y the comustion eventually is not added to the system, and the isentropic efficiencies of expansion and compression processes. Optimizations presented here are aimed to find analytical solutions for imum power and imum thermal efficiency according to specific conditions of the system such as the temperature of the thermal reservoirs and properties of the working fluid. Results are presented of the influence of physical parameters that affect system performance optimizations for oth proposals. t is hoped that the results otained may provide a more appropriate evaluation of real power cycles, esides eing useful for projects seeking more efficient systems or systems with high power outputs. Keywords: Brayton cycle, optimization, power. NRODUON he Brayton cycle is an ideal model of thermodynamics cycle widely used in the representation of gas power systems, which are used oth in transport application as well as in stationary power generation, and may, in the latter case, e operating together with steam power plants for electricity production, even having the advantage to offer weight and size reduced when compared to steam systems. owever, the utilization of these ideal models started to e questioned due to the existent lack etween them and the real thermal cycles. herefore, as the field of thermodynamic has een developing proposals egan to emerge to make the analysis of these models closer to the operation of real heat engines. ith this, it egan to emerge a new ranch in the thermal area which started to e recognized as finite-time thermodynamics. urzon e Ahlorn (975 proposed a new evaluation method of a reversile arnot cycle ased in the fact that the imum thermal efficiency of a heat engine operating etween two thermal reservoirs with temperatures and, given y would e inconsistent with the operation of real heat engines and, therefore, would e a operation limit too idealized. he authors also paid attention to the fact that for a heat engine achieve such efficiency the temperatures of the working fluid under the heat addition and heat rejection processes should e the same as the temperatures of the reservoirs. For such condition could e achieved the time required for these thermal exchanges would tend to infinity, so producing a finite quantity of work, ut with the cycle operating so slowly that it would make its power output tended to zero. hus, modeling a reversile arnot cycle with a finite temperature difference etween the thermal reservoirs and the working fluid it was found that the cycle efficiency operating under imum power conditions is equal to. For this type of model, where the only irreversiilities are the temperature difference etween the reservoirs and the working fluid, was given the name of endoreversile cycle. Using the same principle, u (988, 99 realized an optimization of an endoreversile arnot cycle and of an endoreversile Brayton cycle developing for oth an analytical expression for the power output and optimizing it numerically. rahim et al. (99 also optimized the power output of arnot and Brayton cycles considering oth the cases where the heat capacities of the reservoirs are infinite as in the case they are finite. u e Kiang (99,99 studied the influence of the non-isentropic compression and expansion in the efficiency of a Brayton cycle operating 057

2 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition under imum power and showed that the inclusion of internal irreversiilities in the endoreversile arnot cycle proposed y urzon e Ahlorn (975 could e done y a single parameter. he study of the effects of thermal resistance, heat leakage and internal irreversiilities in the performance of a heat engine, along with the optimization of the thermal efficiency and the power output, was presented y hen (99 and hen et al. (997. he analysis of an endoreversile Brayton cycle done y hen et al. (00 compared the results otained from the optimization of the power output with those otained y the optimization of the power density, the ratio etween the power output and the imum specific volume of the cycle. Zhang et al. (009 optimized a Brayton cycle coupled with a reverse Brayton cycle considering several types of losses found in real heat engines and Sánchez-Orgaz et al.(00 presented a general model for a multi-step regenerative and irreversile Brayton cycle with reheating etween the turines and intercooling etween the compressors otaining relations for imum power and imum efficiency. Zhang et al. (0 analyzed and optimized the arrangement of a regenerative Brayton cycle parallel to two inverse Brayton cycles where the regeneration process occurs efore the inverse cycles. Analytical expressions for the efficiency and the power output was otained and optimized. n this paper is presents the development, analysis and comparison etween the optimization of the power output and thermal efficiency of a Brayton cycle considering several types of irreversiilities found in real heat engines, such as the thermal resistances arising from the temperature difference etween the cycle and the two thermal reservoirs, heat leakage, in which part of the energy supplied y the comustion eventually is not added to the system, and the isentropic efficiencies of expansion and compression processes.. ERMODYNAM ANAYSS A Brayton cycle with internal irreversiilities operating at steady state etween a reservoir with a high temperature and a reservoir with a low temperature, oth with infinite heat capacities, is showed in Fig. (a. he process - is an isoaric heat addition and the -s an isentropic expansion, while the process - is a non-isentropic expansion considering the irreversile nature of real turines. n the process - there is an isoaric heat rejection and in -s occurs the isentropic compression, eing the process - the non-isentropic compression found in real compressors. n the power plant model of Fig. ( the rate of heat transfer asored y the working fluid at high temperature receives the name of Q, while Q is the rate of heat rejection from the working fluid to the reservoir at low temperature. t is considered the existence of a rate of heat leakage Q etween the two reservoirs, with Q, the total rate of heat transfer provided y the high temperature reservoir, eing the sum of heat transfer rejected to the low temperature reservoir Q the sum of Q plus Q. Q plus Q, and the total rate of (a ( Figure. (a -s diagram of Brayton cycle ( power plant model with heat leakage and temperature difference etween the reservoirs and the heat engine. hus, the total rates of heat transfer provided y the hot reservoir and rejected to the cold reservoir are given, respectively, y: 058

3 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition Q Q Q ( Q Q Q ( Each of the heat transfer processes acting on the working fluid can e also calculated y the effectiveness heat exchangers as follows: of the where Q ( Q ( is the heat capacity rate of the working fluid. he heat leakage rate is given y: Q (5 where the factor is the rate of heat transfer etween the two reservoirs per unit temperature for calculating the heat leakage. he effectiveness of the heat exchangers, considered countercurrents, in the hot side and in the cold side are given, respectively, y: exp( N (6 exp( N (7 with the numer of transfer units N and N eing calculated as: N U A / (8 N U A / (9 where U and U are the overall heat transfer coefficients of the hot and cold reservoirs, respectively. From Eq. ( and Eq. ( yields: (0 ( ( ( From the first law of thermodynamics for cycles, the power output of the cycle is given y: ( Q Q solating from Eq. ( yields: ( Sustitution of Eq. ( into Eq. ( yields: ( ( ( ( he non isentropic efficiencies of the compressor and turine are given, respectively, y: 059

4 Proceedings of the nd nternational ongress of Mechanical Engineering OBEM0 Novemer -7, 0, Rieirão Preto, SP, Brazil s (5 s (6 leading to: s (7 s (8 Sustitution of Eq. ( into Eq. (7 and Eqs. (0 and ( into Eq. (8 yields: ( ( ( s (9 s ( (0 he following relation is otained from the second law of thermodynamics: s s ( he sustitution of Eqs. (0, (, (9 and (0 into Eq. ( finally gives: 0 a a a ( where: ( ( a ( a ( 5 a (5 (6 (7 ( ( ( (8 ( ( ( ( ( ( (9 5 ( ( (0 hus, there was otained a second degree polynomial for the power output of the Brayton cycle as a function of the temperature of a single state of the cycle, which is the compressor exit temperature : SSN

5 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition a a 5 ( a a a a a. POER OUPU OPMZAON he imization of as a function of can e done through the operation 0 utilizing Eq. ( to otain the following expression: where: 0 ( a ( ( a 7 8 a 5 (5 he optimum temperature for the power output imization is then given y: , 6 (6 where and the optimum temperatures,,, and, for the cycle to operate at imum power can e calculated sustituting Eq. (6 into Eqs. (, (0, ( e (. ikewise, the rates of heat transfer can e otained sustituting the values of the optimum temperatures into Eqs. ( to (. astly, expressions for the entropy generation rate S g, and for the thermal efficiency at imum power condition are written as: S Q Q Q Q Q Q,,,, g (7, (8 Q Q Q,,. ERMA EFFENY OPMZAON he thermal efficiency can e written as: Q (9 9 0 where: (0 9 ( 0 he imization of is otained in a similar manner to the power output imization sustituting Eq. ( into Eq. (9 and making 0. his operation yields to: 05

6 SSN Proceedings of the nd nternational ongress of Mechanical Engineering OBEM0 Novemer -7, 0, Rieirão Preto, SP, Brazil where: 0 ( 0 a 0 a a ( 0 a 9 0 8a a 7 a ( 6 a a a (5 9 he optimum temperature for the efficiency is then otained:, (6 n a similar manner to the power imization, Eq. (6 can e used to evaluate the optimum power output, Eq. (, the optimum temperatures of the cycle, Eqs. (0, ( and (, and the rates of heat transfer, Eqs. ( a (. Expressions for the optimum entropy generation rate at imum thermal efficiency condition thermal efficiency are finally written as: S g, e for the imum S Q,,,, g, (7 Q Q Q Q Q (8 Q Q Q,, 5. RESUS AND DSUSSON he relation etween the dimensionless power output ( and the thermal efficiency is showed in Fig. to Fig. 5 varying different characteristic parameters of the cycle. From these graphs it is noticed that there is a single point where the power output is at its imum and also a single point where the thermal efficiency is imum, and in this latter case the power is slightly less than the imum power output. Attention should e paid to the fact that the optimal region of operation is located etween these two extreme points, ecause the operation points situated in the left of provide lower values oth in power output as well as in thermal efficiency than the points operating in the optimal region, which also ends up happening at points elow. herefore, it is in the region to the right of and aove, which is the optimal region of cycle operation, where the operation points show the highest values of power and efficiency Fig. presents the relation ( for some values of the reservoir temperature ratio, showing that the increase of this factor leads to a considerale addition oth in as well as in. he influence of N and N in the and values is shower in Fig., making clear the influence of these parameters in the cycle ( performance. he graphic also shows that the increase of N and N, when their values are closer to the unity, leads to a great increase in and in. owever, this ecomes less evident each time this increase happens, making the improvement in the performance of the cycle less apparent for posterior additions in the values of these parameters. Fig. indicates the strong influence of the isentropic efficiencies in the imum values that the cycle can achieve with respect to the power output and thermal efficiency, and that the increase in leads to etter results than the increase in. he heat leakage is also analyzed in Fig 5, where it can e oserved that the value of is independent of and this parameter has only influence on the cycle efficiency. 05

7 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition Figure. Variation of power output y the thermal efficiency for various values of / for constants N, N 0.9 and 0.0. c Figure. Variation of power output y the thermal efficiency for various values of /, 0. 9 and c 0.0. N and N for constants 05

8 SSN Proceedings of the nd nternational ongress of Mechanical Engineering OBEM0 Novemer -7, 0, Rieirão Preto, SP, Brazil Figure. Variation of power output y the thermal efficiency for various values of and for constants /, c N N and 0.0. Figure 5. Variation of power output y the thermal efficiency for various values of N N and 0.9. c for constants /, 05

9 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition (a ( Figure 6. Ratio of the power output at imum efficiency to the imum power output with respect to (a / and ( N (when required the selected constants are /, N N, 0. 9 and 0.0. c he ehavior of the ratio of the power output at imum efficiency to the imum power output,, can e seen in Figs. 6(a and 6( according to / and N, respectively. approaches the unity for lower values of N, N, and c and approaches 85% when theses parameters are increased. he difference etween the powers under these two conditions is also highlighted when increased. / 055

10 SSN Proceedings of the nd nternational ongress of Mechanical Engineering OBEM0 Novemer -7, 0, Rieirão Preto, SP, Brazil (a ( Figure 7. Ratio of the entropy generation rate at imum efficiency to the entropy generation rate at imum power output with respect to (a / e ( N (when required the selected constants are /, N, N 0.9 and 0.0. c he ratio of the entropy generation rate at imum efficiency to the entropy generation rate at imum power, S, can e seen in Figs. 7(a and 7( according to / and N, respectively. Similarly to, the g, S g, difference etween the performance of oth optimizations ecomes more evident for higher values of and N, N,, /. owever, in this case, this difference comes close to 70% when these parameters suffer an increase. his c 056

11 SSN Vitor Pereira Repinaldo, Santiago del Rio Oliveira, Vicente uiz Scalon Optimization of a Brayton ycle at Maximum Power Operating ondition and at Maximum hermal Efficiency Operating ondition makes clear the aility of the thermal efficiency optimization to significantly reduce the value of S, g although with a small reduction in the power output as seen in Figs. 6(a and 6(. (a ( Figure 8. Ratio of the imum thermal efficiency to the thermal efficiency at imum power output with respect to (a / e ( N (when required the selected constants are, N, N c 0. 9 and 0.0. astly, the ratio of the imum thermal efficiency to the thermal efficiency at imum power,, can e seen in Figs. 8(a and 8( as function of respect to the parameters N, N,, and c / and N, respectively. t can e oserved the same ehavior with / of Figs. and. t is noticed that the imum efficiency proved 057

12 SSN Proceedings of the nd nternational ongress of Mechanical Engineering OBEM0 Novemer -7, 0, Rieirão Preto, SP, Brazil e ale to reach a value approximately 0% greater than the efficiency of the cycle operating at imum power, justifying the considerale loss in the power output indicated y the value 0, 85 in Fig. 6(a e 6( under the same conditions. 6. ONUSONS he operation of Brayton cycle under imum efficiency condition proved ale to produce lower values of entropy generation rate and greater values of thermal efficiency, despite the considerale loss of power, when compared to a cycle operating at imum power. herefore, the ideal type of operation will always depend on the purpose of the project, which may seek greater efficiencies and lower entropy generation rates at the expense of greater power output, or get lower efficiencies and greater entropy generation rates aiming higher powers. owever, what cannot e forgotten is that it will not e a single optimal point ideal for all cases, ut a range of optimum operation that will provide the est performances for the cycle, eing situated etween the two optimal conditions presented in this paper. hus, when it is wanted a greater power output the cycle must operate in the left part of this range closer to the point of imum power leaving the region to the right when greater efficiencies are aimed. An example of the search for intermediate operating points within this optimum range can e found in works that propose the ecological optimization, where the goal is to find an intermediate point which represents the est compromise etween a high power output and a low entropy generation rate. 7. REFERENES hen, J., 99. he imum power output and imum efficiency of an irreversile arnot heat engine. Journal of Physics D: Applied Physics, Vol. 7, p.. hen,., Sun, F. and u,., 997. nfluence of internal heat leak on the power versus efficiency characteristics of heat engines. Energy onversion and Management, Vol. 8, p 50. hen,., Zheng, J., Sun, F. and u,., 00. Performance comparison of an endoreversile closed variale temperature heat reservoir Brayton cycle under imum power density and imum power conditions. Energy onversion and Management, Vol., p.. urzon, F.. and Ahlorn, B., 975. Efficiency of a arnot engine at imum power output. American Journal of Physics, Vol., p.. rahim, O. M., Klein, S. A. and Mitchell, J.., 99. Optimum heat power cycles for specified oundary conditions. ASME Journal of Engineering for Gas urines and Power, Vol., p 5. Sánchez-Orgaz, S., Medina, A. and ernández, A.., 00. hermodynamic model and optimization of a multi-step irreversile Brayton cycle. Energy onversion and Management, Vol. 5, p.. u,., 988. Power optimization of a finite-time arnot heat engine. Energy, Vol., p. 68. u,., 99. Power optimization of a endoreversile Brayton gas heat engine. Energy onversion and Management, Vol., p. 56. u,. and Kiang, R.., 99. Power performance of a nonisentropic Brayton cycle. ASME Journal of Engineering for Gas urines and Power, Vol., p. 50. u,. and Kiang, R.., 99. Finite-time thermodynamic analysis of a arnot engine with internal irreversiility. Energy, Vol. 7, p. 7. Zhang,., hen,. and Sun, F., 009. Power and efficiency optimization for comined Brayton and inverse Brayton cycles. Applied hermal Engineering, Vol. 9, p Zhang,., hen,. and Sun, F., 0. Energy performance optimization of comined Brayton and two parallel inverse Brayton cycles with regeneration efore the inverse cycles. Scientia ranica, Vol. 9, p RESPONSBY NOE he authors are the only responsile for the printed material included in this paper. 058

OPTIMIZATION OF AN IRREVERSIBLE OTTO AND DIESEL CYCLES BASED ON ECOLOGICAL FUNCTION. Paraná Federal Institute, Jacarezinho, Paraná, Brasil.

OPTIMIZATION OF AN IRREVERSIBLE OTTO AND DIESEL CYCLES BASED ON ECOLOGICAL FUNCTION. Paraná Federal Institute, Jacarezinho, Paraná, Brasil. OPIMIZAION OF AN IRREVERSIBE OO AND DIESE CYCES BASED ON ECOOGICA FUNCION André. S. MOSCAO *, Santiago D. R. OIVEIRA, Vicente. SCAON, Alcides PADIA * Paraná Federal Institute, Jacarezinho, Paraná, Brasil.