- Email: [email protected]
Effect of mixed heat-resistances on the optimal configuration and performance of a heat-engine cycle
APPLIED ENERGY
Applied Energy 83 (2006) 537–544
www.elsevier.com/locate/apenergy
Effect of mixed heat-resistances on the optimal configuration and performance of a heat-engine cycle Lingen Chen
a,*
, Xiaoqin Zhu
a,b
, Fengrui Sun a, Chih Wu
c
a
c
Postgraduate School, Naval University of Engineering, Wuhan 430033, PR China b Jiangsu Technical Normal College, Changzhou 223001, PR China Mechanical Engineering Department, US Naval Academy Annapolis, MD 21402, USA Accepted 16 May 2005 Available online 15 August 2005
Abstract The finite-time thermodynamic performance of a generalized Carnot-cycle, under the condition of mixed heat-resistances, is studied. The optimal configuration and the fundamental optimal relation between power and efficiency of the cycle are derived. The results provide some guidance for the design of practical engines. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Heat engine; Optimal configuration; Finite heat-reservoir; Optimal performance; Power; Efficiency
1. Introduction In the analysis of finite-time thermodynamics or entropy-generation minimization [1–8], the basic thermodynamic model is the so-called Ôendoreversible CarnotengineÕ, in which only the irreversibility of the finite-rate heat transfer is considered. *
Corresponding author. Tel.: +1 86 27 83615046; fax: +1 86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2005.05.005
538
L. Chen et al. / Applied Energy 83 (2006) 537–544
A major objective of finite-time thermodynamics is to understand irreversible finitetime processes and to establish the general, natural bounds upon the efficiency or maximum work-output for such processes. Another primary goal of finite-time thermodynamics is to establish general operating principles (e.g., the path that the system should follow for maximum efficiency, work or chemical efficiency) for the system which serves as a model for real processes. It is often the case, in practice, that power is generated from heat, which is carried by a finite amount of materiel with a finite heat-capacity, rather than from heat extracted from an isothermal, infinite reservoir. In the reversible (infinite-time) limit, the cycle, which extracts the maximum work from a finite heat-source, is qualitatively different from the Carnot cycle, and its theoretical efficiency is considerably smaller [9]. For the endoreversible cycle, the research into the effect of having a finite heat-reservoir on the performance includes two aspects: the first is to determine the optimal performance of the given finite thermal-capacity cycle, such as the Carnot cycle [10,11], Rankine cycle [12] and Brayton cycle [13–15]. The optimization of the first aspect may be carried out with fixed heat-input [10] or with variable heat-input [11–15]. The second is to determine the optimal configuration of heat engines under given conditions. For example, the optimal configuration of an endoreversible constant-temperature heat-reservoir heat-engine is the Curzon–Ahlborn engine [16]. The optimal configuration of a NewtonÕs law system (i.e., the heat transfer between the working fluid and the heat-reservoirs obeys Qµ(DT)) with a variable-temperature heat-reservoir heat-engine is a generalized endoreversible Carnot engine [17,18] in which the temperatures of the heatreservoirs and the working fluids change exponentially with time and the ratio of the temperatures of the working fluid and the heat-reservoir is a constant. The optimal performance of a generalized Carnot-cycle for another linear heat-transfer law, i.e., Qµ(1/T) was studied in [19]. The optimal performance of a generalized Carnotcycle, assuming the general heat-transfer law Qµ(DT)n, was studied in [20]. The effect of heat leak on the optimal configuration for the NewtonÕs law system heat-engine was studied in [21]. The effect of heat leak on the optimal configuration for another linear heat-transfer law Qµ(1/T) system heat engine was described in [22]. The finite-time optimal performance and finite-time thermoeconomic optimal performance for an endoreversible Carnot-engine and thermoelectric generator, with heat transfer law qµ(DT1), were investigated by Chen et al. [23–25]. These investigations found that the finite nature of the reservoirs is indeed an important feature which must be taken into account in many energy-conversion systems of practical interest, because the optimal cycles and performances are different in systems with finite, non-isothemal reservoirs versus those with infinite isothermal reservoirs. In this paper, the optimal configuration of the heat engine, with mixed heat-resistances, is studied further. It is assumed that the heat source has a finite, constant heat-capacity, the irreversibility arises from heat resistances, and the heat-transfer between the working fluid and the heat-reservoirs obeys a mixed heat-transfer law, i.e., NewtonÕs heat-transfer law Qµ(DT) and/or another linear heat transfer law Qµ(1/T). The fundamental optimal performance between the power output and effi-
L. Chen et al. / Applied Energy 83 (2006) 537–544
539
ciency of the generalized heat-engine is derived in this paper. The results can provide some guidance for the design of practical engines. 2. Heat-engine model The generalized engine-model and its surroundings to be considered in this paper are shown in Fig. 1. The following assumptions are made for this model. The system adapted is a working fluid alternately connected to a hot source with finite heatcapacity and to a cold sink with infinite heat-capacity. The engine operates in a cyclic fashion with a fixed time s allotted for each cycle. The high-temperature heat-source is assumed to have a constant heat-capacity C, its temperature is given by Tx(t), and its initial temperature is given by TH. The cold sink is assumed, for simplicity, to be infinite in size and therefore it has a fixed temperature TL. The heat transfer between the heat source and the warm working-fluid obeys Q1µ(DT m), while the heat transfer between the cold working fluid and the heat sink obeys Q2µ(DT m), where m = ±l. 3. Optimal configuration For the case of m = 1, one has Z s Q1 ¼ K 1 ðtÞ½T x ðtÞ T ðtÞ dt; Z0 s 1 1 Q2 ¼ K 2 ðtÞ dt; T L T ðtÞ 0
ð1Þ ð2Þ
where T(t) is the temperature of the working fluid, Qi(i = l,2) is the heat flux from the ith reservoir to the working fluid and Ki(t)(i = 1,2) is the thermal conductivity for Finite HotSource
C
T x (t )
Q1
Engine
W
Q2
Infinite ColdReservoir Fig. 1. Model of the engine.
TL
540
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heat transfers between the ith reservoir and the working fluid. At t = 0, the working fluid is in contact with the high-temperature heat-source and is separated from the low-temperature heat-sink by an adiabatic boundary. At a later time t1(0 < t1 < s), contact with the heat source is broken, and the working fluid is placed in contact with the heat sink. Therefore, K1(t) and K2(t) can be written as K 1 ; 0 6 t < t1 ; K 1 ðtÞ ¼ ð3Þ 0; t1 6 t < s; 0; 0 6 t < t1 ; ð4Þ K 2 ðtÞ ¼ K 2 ; t1 6 t < s; where K1 and K2 are constants. From the first law of thermodynamics, the power produced is given by Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ W ¼ E þ dt; T L T ðtÞ 0
ð5Þ
where E is the total energy of the working fluid. The total work produced in one cycle of duration s is Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ dt DE; ð6Þ W ¼ T L T ðtÞ 0 where DE is the change in total energy of the working fluid. Since the engine works cyclically, DE over the full cycle must be zero. Hence, the total work done along one full cycle is Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ W ¼ dt. ð7Þ T L T ðtÞ 0 The rate of change of the entropy of the working fluid is S_ ¼ fK 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ½1=T L 1=T ðtÞg=T ðtÞ.
ð8Þ
Since the engine operates cyclically, the change in the entropy of the working fluid over one full-cycle period is zero, Z s 1 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ DS ¼ dt ¼ 0. ð9Þ T L T ðtÞ 0 T ðtÞ Furthermore, since the heat capacity of the hot source is assumed to be constant, one has dQ1 ¼ CdT x ðtÞ.
ð10Þ
Combining Eqs. (1) and (10) gives the constraint equation for the temperature of the high-temperature heat source, i.e., C T_ x ðtÞ þ K 1 ðtÞ½T x ðtÞ T ðtÞ ¼ 0; where T_ x ðtÞ ¼ dT x ðtÞdt.
ð11Þ
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541
The problem now is to determine the optimal configuration of the model cycles in which the maximum work output is obtained in a given cycle time s. Using Eq. (7) and constraint equations (9) and (11), one has the modified Lagrangian 1 1 L ¼ K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ T L T ðtÞ 1 1 1 þ k1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ T L T ðtÞ T ðtÞ þ lfCT x ðtÞ þ K 1 ðtÞ½T x ðtÞ T ðtÞg; where k is the Lagrangian constant, and l(t) is a function of time. The Euler–Lagrange equations are given by oL d oL ¼0 oT ðtÞ dt oT_ ðtÞ and
oL d oL ¼ 0. oT x ðtÞ dt oT_ x ðtÞ
t1 6 t 6 s;
where a and k1 are constants. Combining Eqs. (11), (15) and (16) gives T x ðtÞ ¼ T H evt 06t < t1 ; T x ðtÞ ¼ const.
ð13Þ
ð14Þ
Combining Eqs. (12)–(14) gives: T ðtÞ ¼ aT x ðtÞ; 0 6 t < t1 ; T ðtÞ ¼ 2k1 =ð1=T L 1Þ;
ð12Þ
t1 6 t 6 s;
ð15Þ ð16Þ
ð17Þ
where v is a constant. Equation (17) indicates that the temperatures of the high-temperature heat-source and the working fluid decrease exponentially with time in the time interval {0,t1}, and the ratio of the temperatures of the working fluid and heat source is a constant. For the case of case m = 1, one has Z s 1 1 Q1 ¼ K 1 ðtÞ dt ð18Þ T ðtÞ T x ðtÞ 0 Z s K 2 ðtÞ½T ðtÞ T L dt ð19Þ Q2 ¼ 0
Using the same method as above, one can obtain T x ðtÞ ¼ T H ut 06t < t1 ; T x ðtÞ ¼ const. t1 6 t 6 s;
ð20Þ
where u is a constant. Equation (20) indicates that the temperature of the high-temperature heat-source decreases linearly with time in the time interval {0, t1}.
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4. Fundamental optimal relation Combining the change in the entropy of the working fluid heat-absorbing process dS x ¼ C ln½1 Q1 =ðCT H Þ
ð21Þ
and the condition of internal reversibility, one can introduce an equivalent temperature of the hot reservoir T H ¼ Q1 =dS x ¼ Q=½C lnð1 Q1 =CT H Þ
ð22Þ
and an equivalent temperature of the working fluid in the heat-absorption process T 1 ¼ T 2 Q1 =Q2
ð23Þ
where T2 is the temperature of working fluid in the heat-rejection process. Therefore, one can derive m Q1 ¼ K 1 ðT m H T 1 Þt 1
ð24Þ
m Q2 ¼ K 2 ðT m L T 1 Þðs t 1 Þ
ð25Þ
g ¼ 1 T 2 =T 1
ð26Þ
and
Combining Eqs. (22)–(25) gives P¼
Q1 Q2 ¼ s K
1
1 ðT H T 1 Þ
g . 1g þ K 2 ½T m ½ð1gÞT m L
ð27Þ
1
Taking the derivative of P with respect to T 1 and setting it equal to zero yields P¼
gK 1 ½T H ð1 gÞ T L
; ð28Þ 1þm 1m m ð1 gÞ½T H T L d1m ð1 gÞ þ 2dT H 2 T L2 þ d1þm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where d ¼ K 1 =K 2 . This result indicates that the influence of a finite heat-source heat on the optimal performance of an internal reversible cycle can be expressed by the equivalent T H ,and the T H does not depend on the heat-transfer law. The introduction of the equivalent temperature T H can turn the finite-source into an infinite-source cycle when the optimal performance is discussed. In this case, however, the configuration of the cycle is not the Carnot type because the heat-absorbing process is not an isothermal processes. The temperature of the hot source varies according to the change of time. The optimal configuration is the Carnot type only when C ! 1 (i.e., Tx(t) = TH).
5. Conclusion The optimal configuration and performance for a generalized Carnot-cycle under the condition of mixed heat-resistance has been studied. The basic characteristic is quite different for different types of heat resistance. When Q1µDT and Q2µDT1,
L. Chen et al. / Applied Energy 83 (2006) 537–544
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the temperature of the hot reservoir decreases exponentially with time prior to t1, and so does the temperature of the working fluid. While in another case Q1µDT1 and Q2µDT, the temperature of the hot reservoir decreases linearly with the time prior to t1. The fundamental optimal relation between power and efficiency of the cycle has been derived. The introduction of the equivalent temperature T H can turn the finite-source into an infinite-source cycle. The results provide some guidance for the design of practical engines.
Acknowledgments This paper is supported by the Program for New Century Excellent Talents in the Universities of PR China and The Foundation for the Authors of Nationally-Excellent Doctoral Dissertations of PR China (Project No. 200136).
References [1] Novikov II. The efficiency of atomic power stations: a review. Atommaya Energiya 1957;3(11):409; Chambadal P. Les Centrales Nucleaires. Paris: Armand Colin; 1957. p. 41–58. [2] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power-output. Am J Phys 1975;43(1):22–4. [3] Andresen B. Finite-time thermodynamics. Physics Laboratory, University of Copenhagen; 1983. [4] Bejan A. Entropy-generation minimization. Boca Raton: CRC Press; 1996. [5] Berry RS, Kazakov VA, Sieniutycz S, Szwast Z, Tsirlin AM. Thermodynamic optimization of finitetime processes. Chichester: Wiley; 1999. [6] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization of entropy-generation minimization of energy systems. J Non-Equilib Thermodyn 1999;24(4):327–59. [7] Wu C, Chen L, Chen J, editors. Recent advances in finite-time thermodynamics. New York: Nova Science Publishers; 1999. [8] Chen L. Finite-time thermodynamic analysis of irreversible processes and cycles. Beijing: Higher Education Press; 2005. [9] Ondrechen MJ, Andresen B, Mozurkewich M, Berry RS. Maximum work from a finite reservoir by sequential Carnot cycles. Am J Phys 1981;49(7):681–5. [10] Yan Z. Thermal efficiency of a Carnot engine at the maximum power-output with a finite thermalcapacity heat reservoir. Chin J Eng Thermophys 1984;5(2):125–31 [in Chinese]. [11] Grazzini G. Work from irreversible heat engines. Energy, The Int J 1991;16(4):747–55. [12] Lee WY, Kin SS. An analytical formula for the estimation a Rankine cycleÕs heat-engine efficiency at maximum power. Int J Energy Res 1991;15(3):149–59. [13] Ibrahim OM, Klein SA, Mitchell JW. Optimum heat-power cycles for specified boundary conditions. Trans ASME J Eng Gas-Turbine Power 1991;113(4):514–21. [14] Chen L, Ni N, Cheng G, Sun F, Wu C. Performance analysis for a real closed regenerated Braytoncycle via methods of finite-time thermodynamics. Int J Ambient Energy 1999;20(2):95–104. [15] Chen L, Zheng J, Sun F, Wu C. Power-density analysis and optimization of a regenerated closed variable-temperature heat-reservoir Brayton cycle. J Phys D: Appl Phys 2001;34(11): 1727–39. [16] Rubin MH. Optimal configuration of a class of irreversible heat-engines I. Phys Rev A 1979;19(3):1272–6. [17] Ondrechen MJ, Rubin MH, Band YB. The generalized Carnot cycles: a working fluid operating in finite-time between finite heat-sources and sinks. J Chem Phys 1983;78(7):4721–7.
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[18] Yan Z, Chen L. Optimal performance of an endoreversible cycle operating between a heat source and sink of finite capacities. J Phys A: Math Gen 1997;30(23):8119–27. [19] Yan Z, Chen J. Optimal performance of a generalized Carnot-cycle for another linear heat-transfer law. J Chem Phys 1990;92(3):1994–8. [20] Chen L, Zhu X, Sun F, Wu C. Optimal configurations and performance for a generalized Carnotcycle assuming the heat-transfer law Qµ (DT)m. Appl Energy 2004;78(3):305–13. [21] Chen L, Zhou S, Sun F, Wu C. Optimal configurations and performances of heat-engines with heat leak and finite heat-capacity. Open Syst Inform Dyn 2002;9(1):85–96. [22] Chen L, Sun F and Wu C.Optimal configuration of a two-heat-reservoir heat-engine with heat leak and finite thermal-capacity. Appl Energy [in press]. [23] Chen L, Sun F, Wu C. Thermoelectric generator with linear phenomenological heat-transfer law. Appl Energy 2005;81(4):358–64. [24] Qin X, Chen L, Sun F, Wu C. Frequency-dependent performance of an endoreversible Carnot-engine with linear phenomenological heat-transfer law. Appl Energy 2005;81(4):365–75. [25] Chen L, Sun F, Wu C. Endoreversible thermoeconomics for heat engines. Appl Energy 2005;81(4):388–96.
Applied Energy 83 (2006) 537–544
www.elsevier.com/locate/apenergy
Effect of mixed heat-resistances on the optimal configuration and performance of a heat-engine cycle Lingen Chen
a,*
, Xiaoqin Zhu
a,b
, Fengrui Sun a, Chih Wu
c
a
c
Postgraduate School, Naval University of Engineering, Wuhan 430033, PR China b Jiangsu Technical Normal College, Changzhou 223001, PR China Mechanical Engineering Department, US Naval Academy Annapolis, MD 21402, USA Accepted 16 May 2005 Available online 15 August 2005
Abstract The finite-time thermodynamic performance of a generalized Carnot-cycle, under the condition of mixed heat-resistances, is studied. The optimal configuration and the fundamental optimal relation between power and efficiency of the cycle are derived. The results provide some guidance for the design of practical engines. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Heat engine; Optimal configuration; Finite heat-reservoir; Optimal performance; Power; Efficiency
1. Introduction In the analysis of finite-time thermodynamics or entropy-generation minimization [1–8], the basic thermodynamic model is the so-called Ôendoreversible CarnotengineÕ, in which only the irreversibility of the finite-rate heat transfer is considered. *
Corresponding author. Tel.: +1 86 27 83615046; fax: +1 86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2005.05.005
538
L. Chen et al. / Applied Energy 83 (2006) 537–544
A major objective of finite-time thermodynamics is to understand irreversible finitetime processes and to establish the general, natural bounds upon the efficiency or maximum work-output for such processes. Another primary goal of finite-time thermodynamics is to establish general operating principles (e.g., the path that the system should follow for maximum efficiency, work or chemical efficiency) for the system which serves as a model for real processes. It is often the case, in practice, that power is generated from heat, which is carried by a finite amount of materiel with a finite heat-capacity, rather than from heat extracted from an isothermal, infinite reservoir. In the reversible (infinite-time) limit, the cycle, which extracts the maximum work from a finite heat-source, is qualitatively different from the Carnot cycle, and its theoretical efficiency is considerably smaller [9]. For the endoreversible cycle, the research into the effect of having a finite heat-reservoir on the performance includes two aspects: the first is to determine the optimal performance of the given finite thermal-capacity cycle, such as the Carnot cycle [10,11], Rankine cycle [12] and Brayton cycle [13–15]. The optimization of the first aspect may be carried out with fixed heat-input [10] or with variable heat-input [11–15]. The second is to determine the optimal configuration of heat engines under given conditions. For example, the optimal configuration of an endoreversible constant-temperature heat-reservoir heat-engine is the Curzon–Ahlborn engine [16]. The optimal configuration of a NewtonÕs law system (i.e., the heat transfer between the working fluid and the heat-reservoirs obeys Qµ(DT)) with a variable-temperature heat-reservoir heat-engine is a generalized endoreversible Carnot engine [17,18] in which the temperatures of the heatreservoirs and the working fluids change exponentially with time and the ratio of the temperatures of the working fluid and the heat-reservoir is a constant. The optimal performance of a generalized Carnot-cycle for another linear heat-transfer law, i.e., Qµ(1/T) was studied in [19]. The optimal performance of a generalized Carnotcycle, assuming the general heat-transfer law Qµ(DT)n, was studied in [20]. The effect of heat leak on the optimal configuration for the NewtonÕs law system heat-engine was studied in [21]. The effect of heat leak on the optimal configuration for another linear heat-transfer law Qµ(1/T) system heat engine was described in [22]. The finite-time optimal performance and finite-time thermoeconomic optimal performance for an endoreversible Carnot-engine and thermoelectric generator, with heat transfer law qµ(DT1), were investigated by Chen et al. [23–25]. These investigations found that the finite nature of the reservoirs is indeed an important feature which must be taken into account in many energy-conversion systems of practical interest, because the optimal cycles and performances are different in systems with finite, non-isothemal reservoirs versus those with infinite isothermal reservoirs. In this paper, the optimal configuration of the heat engine, with mixed heat-resistances, is studied further. It is assumed that the heat source has a finite, constant heat-capacity, the irreversibility arises from heat resistances, and the heat-transfer between the working fluid and the heat-reservoirs obeys a mixed heat-transfer law, i.e., NewtonÕs heat-transfer law Qµ(DT) and/or another linear heat transfer law Qµ(1/T). The fundamental optimal performance between the power output and effi-
L. Chen et al. / Applied Energy 83 (2006) 537–544
539
ciency of the generalized heat-engine is derived in this paper. The results can provide some guidance for the design of practical engines. 2. Heat-engine model The generalized engine-model and its surroundings to be considered in this paper are shown in Fig. 1. The following assumptions are made for this model. The system adapted is a working fluid alternately connected to a hot source with finite heatcapacity and to a cold sink with infinite heat-capacity. The engine operates in a cyclic fashion with a fixed time s allotted for each cycle. The high-temperature heat-source is assumed to have a constant heat-capacity C, its temperature is given by Tx(t), and its initial temperature is given by TH. The cold sink is assumed, for simplicity, to be infinite in size and therefore it has a fixed temperature TL. The heat transfer between the heat source and the warm working-fluid obeys Q1µ(DT m), while the heat transfer between the cold working fluid and the heat sink obeys Q2µ(DT m), where m = ±l. 3. Optimal configuration For the case of m = 1, one has Z s Q1 ¼ K 1 ðtÞ½T x ðtÞ T ðtÞ dt; Z0 s 1 1 Q2 ¼ K 2 ðtÞ dt; T L T ðtÞ 0
ð1Þ ð2Þ
where T(t) is the temperature of the working fluid, Qi(i = l,2) is the heat flux from the ith reservoir to the working fluid and Ki(t)(i = 1,2) is the thermal conductivity for Finite HotSource
C
T x (t )
Q1
Engine
W
Q2
Infinite ColdReservoir Fig. 1. Model of the engine.
TL
540
L. Chen et al. / Applied Energy 83 (2006) 537–544
heat transfers between the ith reservoir and the working fluid. At t = 0, the working fluid is in contact with the high-temperature heat-source and is separated from the low-temperature heat-sink by an adiabatic boundary. At a later time t1(0 < t1 < s), contact with the heat source is broken, and the working fluid is placed in contact with the heat sink. Therefore, K1(t) and K2(t) can be written as K 1 ; 0 6 t < t1 ; K 1 ðtÞ ¼ ð3Þ 0; t1 6 t < s; 0; 0 6 t < t1 ; ð4Þ K 2 ðtÞ ¼ K 2 ; t1 6 t < s; where K1 and K2 are constants. From the first law of thermodynamics, the power produced is given by Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ W ¼ E þ dt; T L T ðtÞ 0
ð5Þ
where E is the total energy of the working fluid. The total work produced in one cycle of duration s is Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ dt DE; ð6Þ W ¼ T L T ðtÞ 0 where DE is the change in total energy of the working fluid. Since the engine works cyclically, DE over the full cycle must be zero. Hence, the total work done along one full cycle is Z s 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ W ¼ dt. ð7Þ T L T ðtÞ 0 The rate of change of the entropy of the working fluid is S_ ¼ fK 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ½1=T L 1=T ðtÞg=T ðtÞ.
ð8Þ
Since the engine operates cyclically, the change in the entropy of the working fluid over one full-cycle period is zero, Z s 1 1 1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ DS ¼ dt ¼ 0. ð9Þ T L T ðtÞ 0 T ðtÞ Furthermore, since the heat capacity of the hot source is assumed to be constant, one has dQ1 ¼ CdT x ðtÞ.
ð10Þ
Combining Eqs. (1) and (10) gives the constraint equation for the temperature of the high-temperature heat source, i.e., C T_ x ðtÞ þ K 1 ðtÞ½T x ðtÞ T ðtÞ ¼ 0; where T_ x ðtÞ ¼ dT x ðtÞdt.
ð11Þ
L. Chen et al. / Applied Energy 83 (2006) 537–544
541
The problem now is to determine the optimal configuration of the model cycles in which the maximum work output is obtained in a given cycle time s. Using Eq. (7) and constraint equations (9) and (11), one has the modified Lagrangian 1 1 L ¼ K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ T L T ðtÞ 1 1 1 þ k1 K 1 ðtÞ½T x ðtÞ T ðtÞ K 2 ðtÞ T L T ðtÞ T ðtÞ þ lfCT x ðtÞ þ K 1 ðtÞ½T x ðtÞ T ðtÞg; where k is the Lagrangian constant, and l(t) is a function of time. The Euler–Lagrange equations are given by oL d oL ¼0 oT ðtÞ dt oT_ ðtÞ and
oL d oL ¼ 0. oT x ðtÞ dt oT_ x ðtÞ
t1 6 t 6 s;
where a and k1 are constants. Combining Eqs. (11), (15) and (16) gives T x ðtÞ ¼ T H evt 06t < t1 ; T x ðtÞ ¼ const.
ð13Þ
ð14Þ
Combining Eqs. (12)–(14) gives: T ðtÞ ¼ aT x ðtÞ; 0 6 t < t1 ; T ðtÞ ¼ 2k1 =ð1=T L 1Þ;
ð12Þ
t1 6 t 6 s;
ð15Þ ð16Þ
ð17Þ
where v is a constant. Equation (17) indicates that the temperatures of the high-temperature heat-source and the working fluid decrease exponentially with time in the time interval {0,t1}, and the ratio of the temperatures of the working fluid and heat source is a constant. For the case of case m = 1, one has Z s 1 1 Q1 ¼ K 1 ðtÞ dt ð18Þ T ðtÞ T x ðtÞ 0 Z s K 2 ðtÞ½T ðtÞ T L dt ð19Þ Q2 ¼ 0
Using the same method as above, one can obtain T x ðtÞ ¼ T H ut 06t < t1 ; T x ðtÞ ¼ const. t1 6 t 6 s;
ð20Þ
where u is a constant. Equation (20) indicates that the temperature of the high-temperature heat-source decreases linearly with time in the time interval {0, t1}.
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L. Chen et al. / Applied Energy 83 (2006) 537–544
4. Fundamental optimal relation Combining the change in the entropy of the working fluid heat-absorbing process dS x ¼ C ln½1 Q1 =ðCT H Þ
ð21Þ
and the condition of internal reversibility, one can introduce an equivalent temperature of the hot reservoir T H ¼ Q1 =dS x ¼ Q=½C lnð1 Q1 =CT H Þ
ð22Þ
and an equivalent temperature of the working fluid in the heat-absorption process T 1 ¼ T 2 Q1 =Q2
ð23Þ
where T2 is the temperature of working fluid in the heat-rejection process. Therefore, one can derive m Q1 ¼ K 1 ðT m H T 1 Þt 1
ð24Þ
m Q2 ¼ K 2 ðT m L T 1 Þðs t 1 Þ
ð25Þ
g ¼ 1 T 2 =T 1
ð26Þ
and
Combining Eqs. (22)–(25) gives P¼
Q1 Q2 ¼ s K
1
1 ðT H T 1 Þ
g . 1g þ K 2 ½T m ½ð1gÞT m L
ð27Þ
1
Taking the derivative of P with respect to T 1 and setting it equal to zero yields P¼
gK 1 ½T H ð1 gÞ T L
; ð28Þ 1þm 1m m ð1 gÞ½T H T L d1m ð1 gÞ þ 2dT H 2 T L2 þ d1þm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where d ¼ K 1 =K 2 . This result indicates that the influence of a finite heat-source heat on the optimal performance of an internal reversible cycle can be expressed by the equivalent T H ,and the T H does not depend on the heat-transfer law. The introduction of the equivalent temperature T H can turn the finite-source into an infinite-source cycle when the optimal performance is discussed. In this case, however, the configuration of the cycle is not the Carnot type because the heat-absorbing process is not an isothermal processes. The temperature of the hot source varies according to the change of time. The optimal configuration is the Carnot type only when C ! 1 (i.e., Tx(t) = TH).
5. Conclusion The optimal configuration and performance for a generalized Carnot-cycle under the condition of mixed heat-resistance has been studied. The basic characteristic is quite different for different types of heat resistance. When Q1µDT and Q2µDT1,
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the temperature of the hot reservoir decreases exponentially with time prior to t1, and so does the temperature of the working fluid. While in another case Q1µDT1 and Q2µDT, the temperature of the hot reservoir decreases linearly with the time prior to t1. The fundamental optimal relation between power and efficiency of the cycle has been derived. The introduction of the equivalent temperature T H can turn the finite-source into an infinite-source cycle. The results provide some guidance for the design of practical engines.
Acknowledgments This paper is supported by the Program for New Century Excellent Talents in the Universities of PR China and The Foundation for the Authors of Nationally-Excellent Doctoral Dissertations of PR China (Project No. 200136).
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