- Email: [email protected]
Thermo-economics for endoreversible heat-engines
APPLIED ENERGY
Applied Energy 81 (2005) 388–396
www.elsevier.com/locate/apenergy
Thermo-economics for endoreversible heat-engines Lingen Chen b
a,*
, Fengrui Sun a, Chih Wu
b
a Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA
Available online 5 November 2004
Abstract The thermoeconomics of endoreversible heat engines has been studied based on the linear phenomenological heat-transfer law [i.e., the heat flux Q µ D(1/T), where T is the absolute temperature]. Analytical formulae for profit, the maximum profit and the corresponding efficiency are derived. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Endoreversible thermodynamics; Endoreversible thermo-economics; Linear phenomenological heat-transfer law; Optimization
1. Introduction The efficiency bounds of heat engines operating between two heat-reservoirs T1 and T2 are given by classical thermodynamics, i.e. the well-known Carnot efficiency gC = 1 T2/T1, and it leads to no power output. The engine efficiency at the maximum power-output, gCA = 1 (T2/T1)0.5, was derived by Novikov [1], Chambadal [2], and Curzon and Ahlborn [3], and is different from the Carnot efficiency. Finite-time thermodynamics has been developed subsequently. Today, finite-time ther-
*
Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2004.09.008
L. Chen et al. / Applied Energy 81 (2005) 388–396
389
modynamics is a large and active field, and more than 2000 related publications have been published. See the recent review papers [4,5]. The main objective of the early research of the finite-time thermodynamics was to optimize the efficiency and power output. Besides the power and efficiency objectives, Salamon and Nitzan [6] investigated the optimal performance of an endoreversible Carnot-engine by taking the exergy efficiency (effectiveness), exergy loss, and profit as optimization objectives. On the basis of SalamonÕs work [6] in 1990, Chen et al. provided the finite-time exergoeconomic analysis method, using the combination of finite-time thermodynamics and thermo-economics [7–10], and derived the finite-time exergoeconomic performance bounds, optimal relations and parameter optimization criteria [11–13]. A similar idea was provided by Ibrahim et al. [14], De Vos [15–17] and Bejan [18]. De Vos [15–17] made use of the basic idea of finite-time thermodynamics in thermo-economics for heat engines in which the heat transfer between the working fluid and the heat reservoirs obey NewtonÕs (linear) heat-transfer law, to derive the relation between the optimal efficiency and economic returns, and analyzed the effect of running costs upon the optimal efficiency with a fixed temperature-ratio between the high- and low-temperature reservoirs. The heat-engineÕs performance is affected by the heat-transfer law [19–21]. On the basis of [15], this paper investigates the thermo-economic performance of an endoreversible heat engine assuming that the linear phenomenological heat-transfer law applies [21]. It can provide some guidance for the performance optimization of thermodynamic systems.
2. Model and analysis Fig. 1 is a sketch of a Novikov power-plant [1], where k is the heat-transfer coefficient, T1 and T2 are temperatures of the high- and low-temperature heat reservoirs, respectively, W is the power output, and T3 is the temperature of the working fluid which absorbs the heat from the high-temperature heat-source. Assuming that the heat-flux rate Q between the heat source and the working fluid obeys the linear phenomenological heat-transfer law, then Q ¼ kð1=T 3 1=T 1 Þ:
ð1Þ
Power output is W ¼ ð1 T 1 =T 3 ÞQ:
ð2Þ
Combing Eqs. (1) and (2) gives W ¼ kð1 T 1 =T 3 Þð1=T 3 1=T 1 Þ:
ð3Þ
Obviously, both the heat-flux rate Q and power output W are functions of the temperature T3, see Fig. 2. There are two zero points in the W–T3 curve: one is at T3 = T2: that means there is no temperature difference between the high- and lowtemperature side of the heat engine and there is no power output; the another is at T3 = T1, which means there are no heat transfers between the heat source and
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Fig. 1. Sketch of a Novikov engine.
engine, and the heat flux is zero. There is a maximum power-output between the two points. Taking the derivative of W with respect to T3 and setting it equal to zero (dW/dT3 = 0) gives T 3 ¼ 2T 1 T 2 =ðT 1 þ T 2 Þ:
ð4Þ
The maximum power output is W max ¼ 0:5kðT 1 T 2 Þ2 =ðT 1 þ T 2 Þ:
ð5Þ
The efficiency of engine is defined as g ¼ W =Q ¼ 1 T 2 =T 3 :
ð6Þ
The efficiency at maximum power-output is gW ¼ 0:5ð1 T 2 =T 1 Þ ¼ 0:5gC :
ð7Þ
The engine works as a true heat-engine when the range of T3 is T2 6 T3 6 T1, whereas for T3 < T2, it functions as a refrigerator and for T3 > T1 as a heat pump. The maximum amount of heat that the realistic engine can convert is given by
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391
Fig. 2. Heat flux Q and power W as functions of temperature T3: (a) heat consumption; (b) power output.
Qmax ¼ QðT 2 Þ ¼ kð1=T 2 1=T 1 Þ:
ð8Þ
The optimal efficiency lies somewhere between the maximum power-output point and the maximum-efficiency point. Thus, the optimal T3 satisfies 2T 1 T 2 =ðT 1 þ T 2 Þ 6 ðT 3 Þopt < T 1 :
ð9Þ
Therefore, 0:5ð1 T 2 =T 1 Þ 6 gW < 1 T 2 =T 1 :
ð10Þ
The total running cost C of the power plant includes two parts: the capital cost that is assumed to be proportional to the investment and, therefore, proportional to the size of the power plant; and a fuel cost that is proportional to the fuel consumption and, therefore, to the heat production Q. Assuming that Qmax is an appropriate measure for the size of power plant (i.e. the capital cost), then the total cost of a power plant is given by [15]
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C ¼ aQmax þ bQ;
ð11Þ
where a and b are the costs per unit heat. According to [15], the profits are defined as q ¼ W =C;
ð12Þ
where q is the power output per unit cost. The objective of this paper is to maximize the profit q. It is a function of temperature T3. Substituting Eq. (11) into (12) yields qðT 3 Þ ¼ W ðT 3 Þ=½aQmax þ bQðT 3 Þ:
ð13Þ
Substituting from Eqs. (1), (3) and (8) into Eq. (13) yields qðT 3 Þ ¼
1 T 1 ðT 3 T 2 ÞðT 1 T 3 Þ ; a T 23 ðT 1 T 2 Þ þ bT 1 T 2 ðT 1 T 3 Þ
ð14Þ
where b = b/a is the price ratio. Taking the derivative of q(T3) with respect to T3 and setting it equal to zero, i.e., (dq(T3)/dT3 = 0) yields ½T 21 ð1 þ bÞT 22 T 23 2T 1 T 2 ½T 1 ð1 þ bÞT 2 T 3 bT 21 T 22 ¼ 0:
ð15Þ
Solving Eq. (15) for T3 yields T 3 ¼ T 1T 2
pffiffiffiffiffiffiffiffiffiffiffi T 1 ð1 þ bÞT 2 þ ðT 1 T 2 Þ 1 þ b : T 21 ð1 þ bÞT 22
ð16Þ
Combining Eqs. (6) and (16) gives the efficiency gq for achieving maximum profits, 1 T 21 ð1 þ bÞT 22 pffiffiffiffiffiffiffiffiffiffiffi : gq ¼ 1 ð17Þ T 1 T 1 ð1 þ bÞT 2 þ ðT 1 T 2 Þ 1 þ b
3. Discussion The effect of b on the relative profit aq versus temperature T3 characteristic of the cycle with T1 = 1000 K and T2 = 300 K is shown in Fig. 3. The value of T3 lies between T1 and T2. When b = 0, the aq–T3 curve is similar to the power curve W–T3. Indeed, if b = 0, q(T3) = W(T3)/(aQmax), that is, the maximum profit q(T3) is obtained at T3 = 2T1T2/(T1 + T2). For b ! 1, the maximum of the curve becomes closer to its reversible point at T3 = T1. The fuel-cost ratio is defined as [15] f ¼
bQ bT 2 ðT 1 T 3 Þ ¼ ; aQmax þ bQ ðT 1 T 2 ÞT 3 þ bT 2 ðT 1 T 3 Þ
i.e. the ratio of the fuel cost to the total cost of the power plant.
ð18Þ
L. Chen et al. / Applied Energy 81 (2005) 388–396
393
Fig. 3. Relative profits aq versus temperature T3 for different values of the price ratio b.
The values of the running cost of different fuels can be seen in Table 1 [15]. In the practical power-plant, the range of f is fixed. Solving Eq. (18) gives the relation between the temperature T3 and price ratio b: T3 ¼
bð1 f ÞT 2 =T 1 T 1: f ð1 T 2 =T 1 Þ þ bð1 f ÞT 2 =T 1
ð19Þ
Substituting from Eq. (19) into Eq. (16) yields 2
b ¼ f ð2 f Þ=ð1 f Þ :
ð20Þ
Substituting from Eq. (20) into Eq. (17) yields the optimal efficiency gq, gq ¼ ð1 T 2 =T 1 Þ=ð2 f Þ:
ð21Þ
The optimal efficiency gq versus fuel cost f characteristic with T1 = 1000 K and T2 = 500 K is shown in Fig. 4. The optimal gq varies smoothly from the point with b = 0 and f = 0, i.e. for energy sources where the investment is the preponderant cost, to the point with b = 1 and f = 1, i.e. for energy sources where the fuel is the predominant cost; and the optimal efficiency gq comes closer and closer to the Carnot efficiency gC. If the temperature ratio T2/T1 is taken as a variable, the optimal efficiency gq curve is shown in Fig. 5. Similarly, if T1 and T2 are given as constants, the optimal efficiency gq increases with the value of f , and if f is given, the optimal Table 1 Relative fuel costs for various energy sources [15] Fuel
f (%)
Renewable Uranium Coal Gas
0 25 35 50
394
L. Chen et al. / Applied Energy 81 (2005) 388–396
Fig. 4. Optimal efficiency gq versus fuel cost f with T2/T1 = 0.5.
Fig. 5. Optimal efficiency gq versus temperature ratio T2/T1 for different values of the fuel cost f: (solid lines) linear phenomenological heat-transfer law; (dotted lines) Newton (linear) heat-transfer law.
efficiency gq decreases linearly with increases of the temperature-ratio T2/T1. This is different from that obtained in [15], in which the relation between the optimal efficiency gq and the temperature ratio T2/T1 is non-linear, as shown by the dotted lines in Fig. 5. In a practical engine, the range of f is 0 6 f 6 0.5, see Table 1 [15]. Therefore, gq is closer to 0.5(1 T2/T1).
4. Conclusion This paper considers the thermo-economics of the Novikov endoreversible engine with the linear phenomenological heat-transfer law. The analytical formula of opti-
L. Chen et al. / Applied Energy 81 (2005) 388–396
395
mal efficiency at maximum profit is derived. The result indicates that suitable fuels are of benefit for optimizing the efficiency and one can maximize the profits for the fixed total-cost C and the cycleÕs heat-reservoir temperature; similarly, if the total cost and fuel cost f are fixed, one can decrease the temperature of the low-temperature heat sink, and achieve greater profit and relative optimal efficiency. The result obtained herein is different from that obtained based on NewtonÕs heat-transfer law [15] and can provide some guidance for research upon the thermo-economics of endoreversible heat engines.
Acknowledgements This paper is supported by the Foundation for the Authors of National Excellent Doctoral Dissertations of the PR China (Project No. 200136) and the National Key Basic Research and Development Program of PR China (Project No. G2000026301).
References [1] Novikov II. The efficiency of atomic-power stations (a review). At Energ 1957;3(11):409. [2] Chambadal P. Les Centrales Nucleaires.. Paris: Armand Colin; 1957. p. 41–58. [3] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power-output. Am J Phys 1975;43(1):22–4. [4] Bejan A. Entropy-generation minimization: the new thermodynamics of finite-size devices and finitetime processes. J Appl Phys 1996;79(3):1191–218. [5] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy-generation minimization of energy systems. J Non-Equilibr Thermodyn 1999;24(4):327–59. [6] Salamon P, Nitzan A. Finite-time optimizations of a NewtonÕs law Carnot-cycle. J Chem Phys 1981;74(6):3546–60. [7] Sieniutycz S, Salamon P, editors. Advances in thermodynamics. Finite-time thermodynamics and thermoeconomics, vol. 4.. New York: Taylor & Francis; 1990. [8] Berry RS, Salamon P, Heal G. On a relation between economic and thermodynamic optima. Resour Energ 1978;1(2):125–37. [9] Clark JA. Thermodynamic optimization: an interface with economic analysis. J Non-Equilibr Thermodyn 1996;11(1/2):85–122. [10] Tsatsaronts G. Thermo-economic analysis and optimization of energy systems. Prog Energy Combust Sci 1993;19(3):227–57. [11] Chen L, Sun F, Chen W. The relation between optimal efficiency and profit rate of a Carnot engine. J Eng Therm Energy Power 1991;6(4):237–40 [in Chinese]. [12] Sun F, Chen L, Chen W. The efficiency and profit holographic spectrum of a two-heat reservoir heatengine. Trans Chin Soc Intern Combust Eng 1991;9(3):286–7 [in Chinese]. [13] Chen L, Sun F, Chen W. Finite-time exergoeconomic performance bounds and optimization criteria for two-heat-reservoir heat-engines. Chin Sci Bull 1991;36(3):233–5 [in Chinese]. [14] Ibrahim OM, Klein SA, Mitchell JW. Effects of irreversibility and economics on the performance of a heat engine. ASME Trans J Solar Energy Eng 1992;114(4):267–71. [15] De Vos A. Endoreversible thermo-economics. Energy Convers Manage 1995;36(1):1–5. [16] De Vos A. Endoreversible economics. Energy Convers Manage 1997;38(4):311–7. [17] De Vos A. Endoreversible thermodynamics of solar-energy conversion.. Oxford: Oxford University Press; 1992.
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[18] Bejan A. Power and refrigeration plants for minimum heat-exchanger inventory. ASME Trans J Energy Resour Technol 1993;115(2):148–50. [19] Wu C, Chen L, Sun F, Chen W. Effect of heat-transfer law on the finite-time exergoeconomic optimal-performance of a Carnot heat-engine. Energy, Int J 1996;21(12):1127–34. [20] De Vos A. Efficiency of some heat-engines at maximum-power conditions. Am J Phys 1985;53(6):570–3. [21] Chen L, Bi Y, Wu C. Unified description of endoreversible cycles for another linear heat-transfer law. Int J Energy Environ Econom 1999;9(2):77–93.
Applied Energy 81 (2005) 388–396
www.elsevier.com/locate/apenergy
Thermo-economics for endoreversible heat-engines Lingen Chen b
a,*
, Fengrui Sun a, Chih Wu
b
a Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA
Available online 5 November 2004
Abstract The thermoeconomics of endoreversible heat engines has been studied based on the linear phenomenological heat-transfer law [i.e., the heat flux Q µ D(1/T), where T is the absolute temperature]. Analytical formulae for profit, the maximum profit and the corresponding efficiency are derived. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Endoreversible thermodynamics; Endoreversible thermo-economics; Linear phenomenological heat-transfer law; Optimization
1. Introduction The efficiency bounds of heat engines operating between two heat-reservoirs T1 and T2 are given by classical thermodynamics, i.e. the well-known Carnot efficiency gC = 1 T2/T1, and it leads to no power output. The engine efficiency at the maximum power-output, gCA = 1 (T2/T1)0.5, was derived by Novikov [1], Chambadal [2], and Curzon and Ahlborn [3], and is different from the Carnot efficiency. Finite-time thermodynamics has been developed subsequently. Today, finite-time ther-
*
Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2004.09.008
L. Chen et al. / Applied Energy 81 (2005) 388–396
389
modynamics is a large and active field, and more than 2000 related publications have been published. See the recent review papers [4,5]. The main objective of the early research of the finite-time thermodynamics was to optimize the efficiency and power output. Besides the power and efficiency objectives, Salamon and Nitzan [6] investigated the optimal performance of an endoreversible Carnot-engine by taking the exergy efficiency (effectiveness), exergy loss, and profit as optimization objectives. On the basis of SalamonÕs work [6] in 1990, Chen et al. provided the finite-time exergoeconomic analysis method, using the combination of finite-time thermodynamics and thermo-economics [7–10], and derived the finite-time exergoeconomic performance bounds, optimal relations and parameter optimization criteria [11–13]. A similar idea was provided by Ibrahim et al. [14], De Vos [15–17] and Bejan [18]. De Vos [15–17] made use of the basic idea of finite-time thermodynamics in thermo-economics for heat engines in which the heat transfer between the working fluid and the heat reservoirs obey NewtonÕs (linear) heat-transfer law, to derive the relation between the optimal efficiency and economic returns, and analyzed the effect of running costs upon the optimal efficiency with a fixed temperature-ratio between the high- and low-temperature reservoirs. The heat-engineÕs performance is affected by the heat-transfer law [19–21]. On the basis of [15], this paper investigates the thermo-economic performance of an endoreversible heat engine assuming that the linear phenomenological heat-transfer law applies [21]. It can provide some guidance for the performance optimization of thermodynamic systems.
2. Model and analysis Fig. 1 is a sketch of a Novikov power-plant [1], where k is the heat-transfer coefficient, T1 and T2 are temperatures of the high- and low-temperature heat reservoirs, respectively, W is the power output, and T3 is the temperature of the working fluid which absorbs the heat from the high-temperature heat-source. Assuming that the heat-flux rate Q between the heat source and the working fluid obeys the linear phenomenological heat-transfer law, then Q ¼ kð1=T 3 1=T 1 Þ:
ð1Þ
Power output is W ¼ ð1 T 1 =T 3 ÞQ:
ð2Þ
Combing Eqs. (1) and (2) gives W ¼ kð1 T 1 =T 3 Þð1=T 3 1=T 1 Þ:
ð3Þ
Obviously, both the heat-flux rate Q and power output W are functions of the temperature T3, see Fig. 2. There are two zero points in the W–T3 curve: one is at T3 = T2: that means there is no temperature difference between the high- and lowtemperature side of the heat engine and there is no power output; the another is at T3 = T1, which means there are no heat transfers between the heat source and
390
L. Chen et al. / Applied Energy 81 (2005) 388–396
Fig. 1. Sketch of a Novikov engine.
engine, and the heat flux is zero. There is a maximum power-output between the two points. Taking the derivative of W with respect to T3 and setting it equal to zero (dW/dT3 = 0) gives T 3 ¼ 2T 1 T 2 =ðT 1 þ T 2 Þ:
ð4Þ
The maximum power output is W max ¼ 0:5kðT 1 T 2 Þ2 =ðT 1 þ T 2 Þ:
ð5Þ
The efficiency of engine is defined as g ¼ W =Q ¼ 1 T 2 =T 3 :
ð6Þ
The efficiency at maximum power-output is gW ¼ 0:5ð1 T 2 =T 1 Þ ¼ 0:5gC :
ð7Þ
The engine works as a true heat-engine when the range of T3 is T2 6 T3 6 T1, whereas for T3 < T2, it functions as a refrigerator and for T3 > T1 as a heat pump. The maximum amount of heat that the realistic engine can convert is given by
L. Chen et al. / Applied Energy 81 (2005) 388–396
391
Fig. 2. Heat flux Q and power W as functions of temperature T3: (a) heat consumption; (b) power output.
Qmax ¼ QðT 2 Þ ¼ kð1=T 2 1=T 1 Þ:
ð8Þ
The optimal efficiency lies somewhere between the maximum power-output point and the maximum-efficiency point. Thus, the optimal T3 satisfies 2T 1 T 2 =ðT 1 þ T 2 Þ 6 ðT 3 Þopt < T 1 :
ð9Þ
Therefore, 0:5ð1 T 2 =T 1 Þ 6 gW < 1 T 2 =T 1 :
ð10Þ
The total running cost C of the power plant includes two parts: the capital cost that is assumed to be proportional to the investment and, therefore, proportional to the size of the power plant; and a fuel cost that is proportional to the fuel consumption and, therefore, to the heat production Q. Assuming that Qmax is an appropriate measure for the size of power plant (i.e. the capital cost), then the total cost of a power plant is given by [15]
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C ¼ aQmax þ bQ;
ð11Þ
where a and b are the costs per unit heat. According to [15], the profits are defined as q ¼ W =C;
ð12Þ
where q is the power output per unit cost. The objective of this paper is to maximize the profit q. It is a function of temperature T3. Substituting Eq. (11) into (12) yields qðT 3 Þ ¼ W ðT 3 Þ=½aQmax þ bQðT 3 Þ:
ð13Þ
Substituting from Eqs. (1), (3) and (8) into Eq. (13) yields qðT 3 Þ ¼
1 T 1 ðT 3 T 2 ÞðT 1 T 3 Þ ; a T 23 ðT 1 T 2 Þ þ bT 1 T 2 ðT 1 T 3 Þ
ð14Þ
where b = b/a is the price ratio. Taking the derivative of q(T3) with respect to T3 and setting it equal to zero, i.e., (dq(T3)/dT3 = 0) yields ½T 21 ð1 þ bÞT 22 T 23 2T 1 T 2 ½T 1 ð1 þ bÞT 2 T 3 bT 21 T 22 ¼ 0:
ð15Þ
Solving Eq. (15) for T3 yields T 3 ¼ T 1T 2
pffiffiffiffiffiffiffiffiffiffiffi T 1 ð1 þ bÞT 2 þ ðT 1 T 2 Þ 1 þ b : T 21 ð1 þ bÞT 22
ð16Þ
Combining Eqs. (6) and (16) gives the efficiency gq for achieving maximum profits, 1 T 21 ð1 þ bÞT 22 pffiffiffiffiffiffiffiffiffiffiffi : gq ¼ 1 ð17Þ T 1 T 1 ð1 þ bÞT 2 þ ðT 1 T 2 Þ 1 þ b
3. Discussion The effect of b on the relative profit aq versus temperature T3 characteristic of the cycle with T1 = 1000 K and T2 = 300 K is shown in Fig. 3. The value of T3 lies between T1 and T2. When b = 0, the aq–T3 curve is similar to the power curve W–T3. Indeed, if b = 0, q(T3) = W(T3)/(aQmax), that is, the maximum profit q(T3) is obtained at T3 = 2T1T2/(T1 + T2). For b ! 1, the maximum of the curve becomes closer to its reversible point at T3 = T1. The fuel-cost ratio is defined as [15] f ¼
bQ bT 2 ðT 1 T 3 Þ ¼ ; aQmax þ bQ ðT 1 T 2 ÞT 3 þ bT 2 ðT 1 T 3 Þ
i.e. the ratio of the fuel cost to the total cost of the power plant.
ð18Þ
L. Chen et al. / Applied Energy 81 (2005) 388–396
393
Fig. 3. Relative profits aq versus temperature T3 for different values of the price ratio b.
The values of the running cost of different fuels can be seen in Table 1 [15]. In the practical power-plant, the range of f is fixed. Solving Eq. (18) gives the relation between the temperature T3 and price ratio b: T3 ¼
bð1 f ÞT 2 =T 1 T 1: f ð1 T 2 =T 1 Þ þ bð1 f ÞT 2 =T 1
ð19Þ
Substituting from Eq. (19) into Eq. (16) yields 2
b ¼ f ð2 f Þ=ð1 f Þ :
ð20Þ
Substituting from Eq. (20) into Eq. (17) yields the optimal efficiency gq, gq ¼ ð1 T 2 =T 1 Þ=ð2 f Þ:
ð21Þ
The optimal efficiency gq versus fuel cost f characteristic with T1 = 1000 K and T2 = 500 K is shown in Fig. 4. The optimal gq varies smoothly from the point with b = 0 and f = 0, i.e. for energy sources where the investment is the preponderant cost, to the point with b = 1 and f = 1, i.e. for energy sources where the fuel is the predominant cost; and the optimal efficiency gq comes closer and closer to the Carnot efficiency gC. If the temperature ratio T2/T1 is taken as a variable, the optimal efficiency gq curve is shown in Fig. 5. Similarly, if T1 and T2 are given as constants, the optimal efficiency gq increases with the value of f , and if f is given, the optimal Table 1 Relative fuel costs for various energy sources [15] Fuel
f (%)
Renewable Uranium Coal Gas
0 25 35 50
394
L. Chen et al. / Applied Energy 81 (2005) 388–396
Fig. 4. Optimal efficiency gq versus fuel cost f with T2/T1 = 0.5.
Fig. 5. Optimal efficiency gq versus temperature ratio T2/T1 for different values of the fuel cost f: (solid lines) linear phenomenological heat-transfer law; (dotted lines) Newton (linear) heat-transfer law.
efficiency gq decreases linearly with increases of the temperature-ratio T2/T1. This is different from that obtained in [15], in which the relation between the optimal efficiency gq and the temperature ratio T2/T1 is non-linear, as shown by the dotted lines in Fig. 5. In a practical engine, the range of f is 0 6 f 6 0.5, see Table 1 [15]. Therefore, gq is closer to 0.5(1 T2/T1).
4. Conclusion This paper considers the thermo-economics of the Novikov endoreversible engine with the linear phenomenological heat-transfer law. The analytical formula of opti-
L. Chen et al. / Applied Energy 81 (2005) 388–396
395
mal efficiency at maximum profit is derived. The result indicates that suitable fuels are of benefit for optimizing the efficiency and one can maximize the profits for the fixed total-cost C and the cycleÕs heat-reservoir temperature; similarly, if the total cost and fuel cost f are fixed, one can decrease the temperature of the low-temperature heat sink, and achieve greater profit and relative optimal efficiency. The result obtained herein is different from that obtained based on NewtonÕs heat-transfer law [15] and can provide some guidance for research upon the thermo-economics of endoreversible heat engines.
Acknowledgements This paper is supported by the Foundation for the Authors of National Excellent Doctoral Dissertations of the PR China (Project No. 200136) and the National Key Basic Research and Development Program of PR China (Project No. G2000026301).
References [1] Novikov II. The efficiency of atomic-power stations (a review). At Energ 1957;3(11):409. [2] Chambadal P. Les Centrales Nucleaires.. Paris: Armand Colin; 1957. p. 41–58. [3] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power-output. Am J Phys 1975;43(1):22–4. [4] Bejan A. Entropy-generation minimization: the new thermodynamics of finite-size devices and finitetime processes. J Appl Phys 1996;79(3):1191–218. [5] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy-generation minimization of energy systems. J Non-Equilibr Thermodyn 1999;24(4):327–59. [6] Salamon P, Nitzan A. Finite-time optimizations of a NewtonÕs law Carnot-cycle. J Chem Phys 1981;74(6):3546–60. [7] Sieniutycz S, Salamon P, editors. Advances in thermodynamics. Finite-time thermodynamics and thermoeconomics, vol. 4.. New York: Taylor & Francis; 1990. [8] Berry RS, Salamon P, Heal G. On a relation between economic and thermodynamic optima. Resour Energ 1978;1(2):125–37. [9] Clark JA. Thermodynamic optimization: an interface with economic analysis. J Non-Equilibr Thermodyn 1996;11(1/2):85–122. [10] Tsatsaronts G. Thermo-economic analysis and optimization of energy systems. Prog Energy Combust Sci 1993;19(3):227–57. [11] Chen L, Sun F, Chen W. The relation between optimal efficiency and profit rate of a Carnot engine. J Eng Therm Energy Power 1991;6(4):237–40 [in Chinese]. [12] Sun F, Chen L, Chen W. The efficiency and profit holographic spectrum of a two-heat reservoir heatengine. Trans Chin Soc Intern Combust Eng 1991;9(3):286–7 [in Chinese]. [13] Chen L, Sun F, Chen W. Finite-time exergoeconomic performance bounds and optimization criteria for two-heat-reservoir heat-engines. Chin Sci Bull 1991;36(3):233–5 [in Chinese]. [14] Ibrahim OM, Klein SA, Mitchell JW. Effects of irreversibility and economics on the performance of a heat engine. ASME Trans J Solar Energy Eng 1992;114(4):267–71. [15] De Vos A. Endoreversible thermo-economics. Energy Convers Manage 1995;36(1):1–5. [16] De Vos A. Endoreversible economics. Energy Convers Manage 1997;38(4):311–7. [17] De Vos A. Endoreversible thermodynamics of solar-energy conversion.. Oxford: Oxford University Press; 1992.
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