Revista Mexicana de Física ISSN: X Sociedad Mexicana de Física A.C. México

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1 Revista Mexicana de Física ISSN: X Sociedad Mexicana de Física A.C. México Aragón González, Gerardo; Canales Palma, Aurelio; León Galicia, Alejandro; Morales-Gómez, J. R. A regenerator can fit into an internally irreversible Brayton cycle when operating in maximum work or efficiency Revista Mexicana de Física, vol. 59, núm., febrero-, 03, pp. -7 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

2 THERMODYNAMICS Revista Mexicana de Física 59 ) 7 FEBRUARY 03 A regenerator can fit into an internally irreversible Brayton cycle when operating in maximum work or efficiency Gerardo Aragón González, Aurelio Canales Palma, Alejandro León Galicia, and J. R. Morales-Gómez PDPA. UAM-Azcapotzalco. Av. San Pablo # 80. Col. Reynosa. Azcapotzalco, 000, D.F. Teléfono y FAX: 55) Received 30 de junio de 0; accepted 5 de agosto de 0 A Brayton cycle with internal irreversibilities due to isentropic efficiencies of the turbine and compressor, is analyzed using two criterions. One was developed by the authors [] to obtain the maximum efficiency with respect to the isentropic temperature ratio. Another was obtained in [] and gives the conditions to obtain a better cycle if a heat exchanger for regeneration is coupled. If the operation regime of the cycle is maximum work or efficiency, then, a regenerator can always be coupled. The conditions of regeneration for other operation regimes are also shown. The numerical optimization of a regenerative cycle is performed. Keywords: Brayton cycle; efficiency; máximum; work. Mediante dos criterios se analiza un ciclo Brayton con irreversibilidades internas debidas a las eficiencias isentrópicas de la turbina y compresor. Uno fue desarrollado por los autores [] para obtener la eficiencia máxima respecto a la razón de temperaturas isoentrópicas. El otro, fue obtenido en [] y proporciona condiciones para obtener un mejor ciclo cuando un intercambiador de calor puede acoplarse como regenerador. Cuando el régimen de operación del ciclo es trabajo máximo o máxima eficiencia, siempre se puede acoplar un regenerador. Se muestran también las condiciones de regeneración para otros regímenes de operación. Se realiza la optimización numérica de un ciclo Brayton regenerativo. Descriptores: Ciclo Brayton; eficiencia; máximo; trabajo. PACS: 040G; 44.60; Introduction A gas turbine is modeled as an isentropic Brayton cycle Figure ) with a working substance that behaves as an ideal gas: pv = RT ) the equation ) was first stated by Clapeyron in 839 [3]. It is also supposed that the specific heat, c p, at constant pressure is constant in all the stages of the cycle. The flow rates of source and sink fluids have to be exactly matched to the circulation rate of the working substance, so that the products of mass rate of flow and specific heat will be equal to ṁc p. Although the components may differ [4], this will be employed as a symbol for these matched products. In Figure, an isentropic Brayton cycle is shown including internal irreversibilities due to the isentropic efficiencies of the turbine and compressor. In the ideal case the thermal efficiency could be written as a pressure rate as follows [5]: γ γ η = εp ) where ε p = p s /p = p 3 /p 4s is the pressure ratio, γ = c p /c v, c p is the constant-volume specific heat and s and 4s are the isentropic outles. Lewins [6] has recognized that the extreme temperatures are subject to limits: a) the environmental temperature and; b) in function of the limits on the adiabatic flame or for metallurgical reasons. The thermal efficiency is maximized without losses, if the pressure ratio grows up to the point that the compressor output temperature reaches its upper limit. These results show that there is no heat transferred in the hot side and as a consequence the work is zero [7]. The limit occurs when the inlet temperature of the compressor equals the inlet temperature of the turbine [8]; as result no heat is added in the heating/combustor; then, the net work is vanished if ε p =. Therefore at some intermediate point the work reaches a maximum and this point is located close to the economical optimum [6]-[8]. In such condition, the outlet temperature of the compressor and the outlet temperature of the turbine are equal T s = T 4s ; Figure ). If this condition is not fulfilled T s T 4s ), it is advisable to couple a heat regeneration in order to improve the efficiency of the system if T s < T 4s [8]. A similar condition is presented when internal irreversibilities due to the isentropic efficiencies of the turbine η ) and compressor η ), are taken into account non isentropic cycle): T < T 4 see Figure and equation 0) of []). Furthermore, if this condition is fulfilled, the following inequality for the pressure ratio is obtained []: ε p < ε p ) max = B + B + η η T η T where B = η ) T + η ) T3 and ε p ) max is the maximum value of the pressure ratio of the cycle analyzed in []. 3)

The thermal efficiency is defined as the relation between the net work and the transferred heat or by the First Law: η = w q H = q L q H = x T T x = x 7) where w, q H and q L correspond to the

3 A REGENERATOR CAN FIT INTO AN INTERNALLY IRREVERSIBLE BRAYTON CYCLE WHEN OPERATING... 3 and from the equation of the isentropic: T 4s = T T s = ps p ) γ γ γ γ = εp = x 6) where x is the isentropic temperature ratio of the cycle. The thermal efficiency is defined as the relation between the net work and the transferred heat or by the First Law: η = w q H = q L q H = x T T x = x 7) where w, q H and q L correspond to the dimensionless work W Q c ) and heats M c ), respectively. The efficiency m T m T ; M=H, L p p of the isentropic cycle can be maximized by the following criterion []. FIGURE. Bayton cycles: isentropic and non isentropic. On the other hand, since the theory of thermodynamic optimization or finite-time thermodynamics FTT), or endoreversible thermodynamics, or entropy generation minimization EGM)) has advanced, Brayton cycles has been also analyzed and optimized for the power, specific power, power density, efficiency, and ecological optimization objectives with the heat transfer irreversibility and/or internal irreversibilities more details are found in [9-0]). In this work, the efficiency to maximum work and the maximum efficiency for the isentropic and non-isentropic cycles are presented. This work also shows an inequality for the isentropic temperature ratio x) which was adapted of the Eq. 3). And if it is applied to maximum work of the non isentropic cycle, then, a regenerator can couple in order to obtain a better cycle. In the section of conclusions a numerical optimization of a regenerative cycle is performed.. Isentropic cycle A Brayton cycle with two coupled reversible counterflow heat exchangers is shown in Fig.. The supposition of heat being reversibly exchanged that is with a vanishingly small temperature difference in a balanced counterflow heat exchanger) is an equivalent idealization to the supposed heat transfer at constant temperature between the working substance of a Carnot or Stirling) isentropic cycle and a reservoir of infinite heat capacity [3]. From Fig., the thermodynamic analysis for the cycle s 3 4s is given by: dh = c p dt = sdt + vdp = sdt + RT p dp 4) For an ideal gas, the Poisson s equation for the adiabatic process is [5]: pv γ = constant 5) Criterion. Let η = w q H = q L qh. Suppose that q H < 0 and q L = 0, for some x. Then, the maximum efficiency η max is given by: η max = w x=x me q. H x=x me = q L x=x me q H x=x me 8) where x me is the value for which the efficiency reaches its maximum. For the isentropic cycle: w = µ x) x ) 9) q H = µ x ; q L = x µ 0) with µ = T. The hypotheses of this criterion are clearly satisfied: q H < 0 and q L = 0 for some x Figure, where E is energy). FIGURE. Heats and net work qualitative behaviors for µ = 0.5. Rev. Mex. Fis. 59 ) 03) 7

4 4 G. ARAGÓN GONZÁLEZ, A. CANALES PALMA, A. LEÓN GALICIA, AND J.R. MORALES-GÓMEZ Thus, the maximum efficiency is given by the Eq. 0): Solving then, η max = x me µ x me = x me µ ) xme µ = µ µ ) x me where w T and w C are the specific work for turbine and compressor respectively. When the work is a maximum, the efficiency is given by: η mw = w C w T 8) Finally, the following additional corollary has been obtained: x me = µ; η max = µ 3) which corresponds to the Carnot efficiency; the other root x me = 0 is ignored. As a result, the work is null for x me = µ, as a consequence the added heat is also null Fig. ). From a historical point of view [5] the result above tells that the maximum efficiency corresponds exactly to the Carnotian efficiency, and it was obtained without Second Law; because of the Eq. 5) was first found by S. Poisson in 83, before of the work of S. Carnot 84), which together with the Eq. ), the Eq. 6) is obtained. Of course, this conclusion is only valid if the working substance has ideal gas behavior. On the other hand, Eq. 5) is a consequence of the Second Law; it is really the equation of an isentropic process. Accordingly to the aforementioned: the best condition for operation of this cycle is maximum work. Figure shows the existence of this maximum. Thus, if Eq. 9) is derived: Corollary 4. For other types of operation regimes is appropriated to fit a heat exchanger between the turbine and compressor outlets as long as the isentropic temperatures fulfill: T 4s T s. 3. Non isentropic cycle By considering the losses effect in the turbine and compressor Figure, cycle ) and taking into account the isentropic efficiencies η, η, respectively []; the temperatures are given by []: then dw dx = 0 = µ x x µ 4) T = T + x ) ; T 4 = η x)) 9) xη x mw = µ and η mw = µ 5) The same result was obtained in [6-8] and []. Moreover, [6] has obtained the following corollaries see also [7]): Thus, that the net work for this cycle same notation for both cycles, using the Eq. 9) and the structure of the Eq. 9)), w net is given by: Corollary. For maximum work, then: w net = η µ x) ) η x 0) T T s = x mw ; T T 4s = T T 4s = x mw x mw = x mw 6) i.e. the temperatures at the outlet of both compressor and turbine are the same. Corollary 3. For maximum work. The work done on the ideal) compressor equals the heat rejected and the heat accepted in the combustor) heat exchanger equals the produce work in the turbine: From the Eq. 0), the following corollaries are easily obtained: Corollary 5. For maximum work: η T 4s = η T s ; or T 4s = IT s ) w C = q L and w T = q H 7) where I = η η. Rev. Mex. Fis. 59 ) 03) 7

5 A REGENERATOR CAN FIT INTO AN INTERNALLY IRREVERSIBLE BRAYTON CYCLE WHEN OPERATING... 5 And, as T 4s T s = I, then: Corollary 6. The inefficiencies in the components increase the non isentropic temperature in the turbine outlet as compared from the corresponding compressor outlet. For maximum work, the efficiency is given by [6]: η mw = η η ) + µη η µη η µ) + µ µη η ) with an extremum value at: x mw = Iµ 3) Similarly, the maximum efficiency is obtained as follows []. It is clear, q H = [ µ + )] x) η x 4) then, the hypothesis from the criterion of the first section are fulfilled the qualitative behavior of Fig. is preserved). Thus the maximum efficiency is given by: η max = w q = x me Iµ and, solving the following cubic equation: 5) x me Iµ = w x me q H xme ) η µ x me ) η x me = ) 6) µ + xme) η x me it is sufficient to obtain the maximum efficiency which is: η max = η η µ µη + ) η µ µ) µ η )+η η )) 7) µ η ) + η ) with an extremun value at: x me = µη + η µ µ) µ η ) + η η )) η µ η ) + η ) 8) The extreme value, x me, is bounded by []: Iµ x me x mw. The behaviors of η max, η mw and η CI = Iµ versus µ, if η = η = 90% are shown in the Fig. 3. FIGURE 3. Behaviour of η max, η mw and η CI = Iµ versus µ, if η = η = 90%. On the other hand, the regenerative criterion Eq. 3)) can be applied. In terms of x: Criterion 7. If with x > x min 9) x min = β + β + 4Iµ 30) ) ) where β = η + I η µ > 0. Then a regenerator can be used to obtain a better cycle. Following [6], only if the temperature of the exhaust working substance leaving the turbine is higher than the exit temperature T 4 > T ) of the working substance in the compressor for the Brayton cycle, a regenerator is used. Otherwise, heat will flow in the reverse direction and decrease the efficiency of the cycle. This point can be directly seen when T 4 < T, because the regenerative rate is smaller than zero and consequently the regenerator does not have a positive role. From Eqs. 9) the following relation is obtained: T = T + x ) < T 4 = η x)) 3) xη which corresponds to a temperature criterion which is equivalent to the inequality 9). Indeed, ) from the Eq. ) 3): x + βx Iµ > 0; with β = η + I η µ > 0 since I > η ). Thus, the minimum value of x results be: x min = β + β + 4Iµ > 0 3) Rev. Mex. Fis. 59 ) 03) 7

6 6 G. ARAGÓN GONZÁLEZ, A. CANALES PALMA, A. LEÓN GALICIA, AND J.R. MORALES-GÓMEZ since β + 4Iµ > β. The other root is ignored because it is negative. Therefore, if x x min, then a regenerator cannot be used. Thus, the inequality 9) is fulfilled. Criterion 8. Applying 36), then: β + 4Iµ η µ µ) η ) µ+η η ) > η µ η ) + η ) 39) If the cycle operates to maximum work or efficiency a counterflow heat exchanger regenerator) between the turbine and compressor outlet is favorable to obtain a better cycle. If the operation regime of the Brayton cycle is maximum work, then, the value x mw of the Eq. 3) fulfills the inequality 9): x mw > x min. Indeed But x mw x min = Iµ + β β + 4Iµ ) Iµ β + β + 4Iµ 4 33) = I Iµ η η ) + µ η )) > 0 34) Now, as the terms between parentheses are positive then: β Iµ + > β + 4Iµ 35) where the following elementary inequality has been applied [3]: 4. Conclusions In this work, if the operation regime is maximum work or efficiency then a regenerator can be coupled to obtain a better cycle. In general, if x satisfies the inequality 9) for a Brayton cycle a regenerator can also be coupled. As illustration, if a regenerator is coupled to the non isentropic Brayton cycle, the efficiency is given by [4]: η η reg x) µ η = x ) ) ε) µ + x xη ) + εη x) 40) where ε corresponds to the effectiveness of the regenerator [7]. For the isentropic cycle, an additional corollary can be obtained: Corollary 9. If the cycle is isentropic then: If a, b > 0, then a < b a < b 36) Therefore, for maximum power the inequality 9) is fulfilled. Now, if the operation regime of Brayton heat engine is maximum efficiency. Also, x me Eq. 8)) satisfies the inequality: x me > x min is fulfilled; because of: η reg max = η mw = µ 4) where η reg max is the maximum efficiency of the cycle with regenerator. In this case, the inequality 9) becomes to equality: x me = x min = µ 4) β + 4Iµ x me x min = η µ + β + ) η µ µ) η ) µ + η η ) η µ η ) + η ) 37) to see that the term betwen parenthesis is positive, we calculate: ) β + 4Iµ η µ µ) η ) µ+η η ) ) η µ η )+η ) = η η µ) + µ) β η µ)+µ) + 4Iµ ) + 4η µ µ) ) > 0 38) FIGURE 4. Behaviour of η mw, η reg max and η max versus µ, if η = η = 90%. Rev. Mex. Fis. 59 ) 03) 7

7 A REGENERATOR CAN FIT INTO AN INTERNALLY IRREVERSIBLE BRAYTON CYCLE WHEN OPERATING... 7 For a non-isentropic Brayton cycle: by optimizing, η reg = η reg x, ε), for some realistic values [5]: η = η = 90%, the behaviors of η mw,ηmax reg and η max versus µ are found see Fig. 4); which fulfill the following inequality: η mw < ηmax reg < η max. The optimization of Eq. 40) gives equations for ε and x which are cumbersome. Albeit, one closed form could be obtained for η reg max. But this is not covered by this work s scope. For maximum efficiency, another regenerator must be coupled [7], since the following inequality: ηmax reg > η max ; should be fulfilled. Further work is underway.. gag@correo.azc.uam.mx. G. Aragón-González, A. Canales-Palma and A. León-Galicia, M. Musharrafie-Martínez, A criterion to maximize the irreversible efficiency in heat engines J. Phys. D: Appl. Phys., ), Y. Zhang, C. Ou, B. Lin and J. Chen, The Regenerative Criteria of an Irreversible Brayton Heat Engine and its General Optimum Performance Characteristics, J. Energy Resour. Technol., 83) 006), 6- DOI:0.5/ H. M. Barkla, The Joule Cycle Eur. J. Phys., 980), H. U. Fuchs, The Dynamics of Heat. st edition, Springer Verlag 996). 5. S. L. Arsenjev, I. B. Lozovitski and Y. P. Sirik The gas equation for a stream 6. J. D. Lewins. The ideal gas Joule at maximum specific work, Proc. Inst. Mech. Engr. 4, 000), H. Cohen and G. F. C. Rogers.Gas Turbine Theory. 4 st edition 4rd edition, Longman 996). 8. R. W. Haywood, Analysis of Engines Cycles. 3 st edition. Pergamon 980). 9. C. Wu, Power optimization of an endoreversible Brayton gas turbine heat engine, Energy Convers. Mgmt., 36) 99) O.M. Ibrahim, S.A. Klein, J.W. Mitchell, Optimum heat power cycles for specified boundary conditions, Trans. ASME J. Engng. Gas Turbine Pow., 3 4) 99), L. Chen, F. Sun, C. Wu, Performance analysis of an irreversible Brayton heat engine, J. Inst. Energy, 70 ) 997), 8.. M. Feidt, Optimization of Brayton cycle engine in contact with fluid thermal capacities, Rev. Gen. Therm., 35 48/49) 996), C. Wu, L. Chen, F. Sun, Performance of a regenerative Brayton heat engine, Energy, ) 996), L. Chen, F. Sun, C. Wu, R.L. Kiang, Theoretical analysis of the performance of a regenerated closed Brayton cycle with internal irreversibilities, Energy Convers. Mgmt., 8 9) 997), L. Chen, N. Ni, G. Cheng, F. Sun, FTT performance of a closed regenerated Brayton cycle coupled to variabletemperature heat reservoirs, Proc. Int. Conf. Marine Engng Nov., 4 8, 996, Shanghai, China. 6. L. Chen, N. Ni, G. Cheng, F. Sun, C. Wu, Performance analysis for a real closed regenerated Brayton cycle via methods of finite time thermodynamics, Int. J. Ambient Energy, 0 ) 999), L. Chen, F. Sun, C. Wu, Effect of heat resistance on the performance of closed gas turbine regenerative cycles, Int. J. Power Energy Syst., 9 ) 999), C.Y. Cheng, C.K. Chen, Power optimization of an irreversible Brayton heat engine, Energy Sources, 9 5) 997), C.Y. Cheng, C.K. Chen, Efficiency optimizations of an irreversible Brayton heat Engine, Trans. ASME, J. Energy Res. Tech., 0 ) 998), C.Y. Cheng, C.K. Chen, Ecological optimization of an endoreversible Brayton cycle, Energy Convers. Mgmt., 39 -) 998), H. S. Leff, Thermal efficiency at maximum work output: New results for old engines, Am. J. Phys., 55 7) 987), M. J. Moran, and H. N. Shapiro. Fundamentals of Engineering Thermodynamic. st edition, John Wiley and Sons Inc. 99). 3. M. J. Cloud and B. C. Drachman. Inequality, With Applications to Engineering. st edition, Springer Verlag, N. Y. 998), J. M. M. Rocco, S. Velasco, A. Medina and A. Calvo Hernández, Optimum performance of a regenerative Brayton thermal cycles, J. Appl. Phys., 8 6) 997), Diesel and Gas turbines. Rev. Mex. Fis. 59 ) 03) 7

Optimization of an irreversible Carnot engine in finite time and finite size

INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 52 4) 309 314 AGOSTO 2006 Optimization of an irreversible Carnot engine in finite time and finite size G. Aragón-González, A. Canales-Palma, A. León-Galicia, and