- Email: [email protected]
Finite-time thermodynamic analysis of a solar driven heat engine
Exergy Int. J. 1(2) (2001) 122–126 www.exergyonline.com
Finite-time thermodynamic analysis of a solar driven heat engine Ahmet Z. Sahin Department of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia
(Received 9 December 1999, accepted 13 September 2000)
Abstract — The collective role of radiation and convection modes of heat transfer in a solar driven heat engine is investigated through a finite time thermodynamics analysis. Heat transfer from hot reservoir is assumed to be radiation and/or convection dominated. The irreversibilities due to these finite rate heat transfers were considered in determining the limits of efficiency and power generation that were discussed through varying process parameters. Results were compared with Curzon–Ahlborn and Carnot analysis cases. It is found that the upper limit of efficiency is a function of both the functional temperature dependence of heat transfer and relevant system parameters. 2001 Éditions scientifiques et médicales Elsevier SAS
1. INTRODUCTION
Nomenclature C Q rC rR rT T W
heat transfer coefficient rate of heat transfer . . = CHC /CLC = CHR TH3 /CLC = TL /TH temperature . . . . . . . rate of work . . . . . .
. . . . . . . W·K−1 or W·K−4 . . . . . . . W
. . . . . . . . . . . . . .
Greek symbols η θ
efficiency = Ta /TH
Subscripts a b C CA H HC HR L LC th
boiler side condenser side Carnot Curzon–Ahlborn high temperature reservoir high temperature side convection high temperature side radiation low temperature reservoir low temperature side convection thermal
E-mail address: [email protected] (A.Z. Sahin).
122
K W
The study of irreversible thermodynamic cycles has been undertaken by many researchers after Curzon and Ahlborn’s [1] work. Efficiency of some heat engines where the power output is limited by the rate of heat supply was studied by De Vos [2] at maximum-power conditions. Wu [3] derived a mathematical expression for the power output of an endo-reversible heat engine. Chen [4] studied the optimum operating temperatures in a solar driven heat engine and the optimum performance of the system. Chen [5], later, introduced a new general cyclic model including heat leaks and dissipative process inside the working fluid to optimize the performance of an irreversible heat engine. Sahin et al. [6] studied the efficiency of a Joule–Brayton engine at maximum power density with a consideration of engine size. Their results show that the efficiency at maximum power density is always greater than that presented by Curzon and Ahlborn [1]. In a later publication, they extended their work [7] to include irreversibilities at the compressor and the turbine of a Joule–Brayton engine along with considerations of pressure losses at the burner for the performance analysis. Medina et al. [8] extended the work of Sahin et al. [6, 7] to a regenerative Joule–Brayton cycle where the optimal operating conditions at the engine were expressed in terms of the compressor and turbine isentropic efficiencies and of the heat exchanger effi 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1164-0235(01)00018-8/FLA
A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
ciency. Cheng and Chen [9] studied the effect of regeneration on power output and thermal efficiency of an endoreversible Brayton cycle. Goktun [10] presented a relation between the design parameter of an internally and externally irreversible Carnotlike solar driven heat engine to find the maximum power and the efficiency at maximum power output. He [11] also investigated the optimum performance of an irreversible heat engine system driven by a corrugated sheet solar collector. Erbay and Yavuz [12] analyzed the endo-reversible heat engine by introducing radiation heat transfer coefficients as a function of temperature with combined heat transfer to obtain the design parameters for the operating under the maximum power conditions. The principle of operation of a solar-thermal power plant is presented by Lund [13] in terms of finite heat transfer rates and an internally reversible heat engine by presenting some parametric equations. Finite time thermodynamics analysis of heat engines is usually restricted to systems having either linear of power law heat transfer dependence to the temperatures of both reservoirs and engine working fluid. However, radiation and convection modes of heat transfer are often coupled and play a collective role in processes of heat engine. In a recent study, Sahin [14] investigated the optimum operating conditions of solar driven heat engines with radiation dominated heat transfer from the high temperature reservoir and convection dominated low temperature reservoir heat transfer. As an extension of Sahin’s work [14], both convection and radiation modes of heat transfer are considered from the high temperature reservoir.
2. ANALYSIS Solar driven heat engines used in the atmosphere operate on the simultaneous convection and radiation modes of heat transfer from the high temperature reservoir. Depending on the environmental and operating conditions either of the modes may be dominant. However, in general, the finite time heat transfer amount from high temperature reservoir can be written as QH = CHR TH4 − Ta4 + CHC (TH − Ta ) (1) On the other hand, convection heat transfer is assumed to be the main mode of heat transfer to low temperature reservoir QL = CLC (Tb − TL )
(2)
Figure 1. Solar driven endo-reversible heat engine.
The work output according to the first law is W = QH − QL = ηth QH
(3)
where η = 1 − Tb /Ta . According to the second law of thermodynamics for the endo-reversible heat engine shown in figure 1 QH QL = Ta Tb
(4)
Using equations (1) and (2) in equation (4), a relationship between Tb and Ta is obtained as CHR TH4 − Ta4 CHC TH − Ta −1 Tb = TL 1 − − (5) CLC Ta CLC Ta The optimum temperatures, Ta and Tb , at the maximum power output condition can be obtained by differentiating equation (3) with respect to Ta and setting it equal to zero. After a lengthy algebra the resultant equation to be solved for Ta is found to be 4rR3 θ 11 + rR2 (9rC + 8)θ 8 − 8rR2 (rC + rR )θ 7 + 2rR 3rC2 + 5rC + 2 θ 5 − rR 2(4 + 5rC )(rC + rR ) + 3rT θ 4 + 4rR (rC + rR )2 θ 3 + rC (1 + rC )2 θ 2 − 2rC (1 + rC )(rC + rR )θ + (rC + rR ) rC (rC + rR ) − rT = 0
(6)
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A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
where θ = Ta /TH , rR = (CHR /CLC )TH3 , rC = CHC /CLC , and rT = TL /TH . The solution of equation (6) can be done numerically. It should be noted that the physically meaningful root of θ in equation (6) is the one located between 0 and 1.
3. DISCUSSION From the solution of equation (6), the effect of the radiative (rR ) and convective (rC ) mode of heat transfers on the optimum temperatures (Ta and Tb ) and on the efficiency at maximum power output (η) can be studied, respectively. Figure 2(a) shows the variation of θ with respect to rR for three different values of rT and rC = 0.1. As radiation (rR ) increases θ increases and approaches to 1. This means that the optimal temperature Ta gets closer to the source temperature TH as radiation increases. For higher values of rT , the ratio of the optimum temperature Ta to the source temperature TH is higher in general. For no radiation case, as rR → 0 (convection mode only), it can be easily found from equation (6) that √ rC + rT θ→ (7) 1 + rC
(a)
This indicates that as rT (i.e., TL ) increases θ (i.e., the optimum value of Ta for fixed value of TH ) increases. Since Ta /TL = θ/rT , using equation (7) √ rC + rT Ta = TL rT (1 + rC ) The optimum value of Tb , on the other hand, becomes √ rC + rT Tb =√ TL rT (1 + rC ) Thus the efficiency for this case approaches the Curzon– Ahlborn efficiency √ η → ηCA = 1 − rT As radiation dominates rR → ∞, it can be shown that, from equation (6), θ 8 − 2θ 4 + 1 = 0 or
124
4 2 θ −1 =0
(b) Figure 2. Solution of θ in equation (6) for three different cases of rT : (a) θ versus rR for rC = 0.1, and (b) θ versus rC for rR = 0.1.
The only meaningful root of this equation is +1, i.e., θ =1 This means that Ta /TL = 1/rT and Tb /TL = 1. Thus the efficiency in this case approaches the Carnot efficiency η → ηC = 1 − rT Figure 2(b) shows the effect of convection (rC ) on the optimum temperature variation θ for fixed value of rT = 0.1 and three different values of rT . θ reaches a
A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
(a)
(a)
(b) (b) Figure 3. Optimum temperature variations of Ta and Tb for the case rT = 0.1: (a) Temperature variations with respect to rR for rC = 0.1; (b) Temperature variations with respect to rC for rR = 0.1.
weak minimum and then increases as rC is increased. Higher values of rT yields higher values of θ . Variation of optimum temperatures, Ta and Tb , versus rR is shown in figure 3(a) where both rC and rT are fixed to be equal to 0.1. As rR approaches 1, the difference between Tb and TL increases. Tb approaches to a constant value as rR increases. The optimum temperature variations with respect to rC are given in figure 3(b) for fixed values of rR = 0.1
Figure 4. Variation of efficiency (η ) for constant value of rT = 0.1 and three cases of rC : (a) Efficiency variation with respect to rR ; (b) Efficiency variation with respect to rC .
and rT = 0.1. Tb /TL increases while Ta /TL initially decreases to a minimum and then increases as rC increases. The gap between Ta and Tb also increases. The variation of efficiency at maximum power output (η) for the present case is given with respect to rR in figure 4(a) for a fixed value of rT = 0.1 and three cases of rC . Since rT is taken to be constant, both Carnot and Curzon–Ahlborn efficiencies, (ηC ) and (ηCA ), are constant and equal to 0.9 and 0.6838, respectively. ηCA is shown in a dotted line in figure 4(a). It is clear that η is higher than ηCA for the whole range of rR values. The difference in these efficiencies is larger for smaller
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A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
rC values. As rR is increased η reaches to a maximum and then decreases. Finally, the effect of convection on the efficiency is given in figure 4(b) for fixed value of rT = 0.1 and three different values of rR . Curzon–Ahlborn efficiency is also included in figure 4(b) in a dotted line for reference. Clearly the efficiency may become much higher than the Curzon–Ahlborn efficiency, especially for low values of rC for which case the radiation becomes the dominant mode of heat transfer from the source temperature TH . As convection (rC ) increases, however, the efficiency approaches to Curzon–Ahlborn efficiency for all values of rR . As a numerical example, consider the case where rT = 0.1, rC = 0.1 and rR = 0.2, the efficiency becomes nearly 73% which is 4.5% higher than the Curzon–Ahlborn efficiency. As rR increases to 0.8 the efficiency decreases to 69.5%. This indicated a decrease of 3.5% as compared to the case of rR = 0.2. On the other hand, when rR decreases to 0 the efficiency increases sharply as shown in figure 4(b). As rC is increased, however, the effect of rR on the efficiency diminishes as seen in figure 4 (a) and (b). As a result, both convection and radiation modes of heat transfer from the high temperature reservoir need to be considered to assess the efficiency accurately, since variation in the either of the modes may influence the efficiency considerably.
4. CONCLUSIONS Radiation and convection modes of heat transfer are often coupled and play a collective role in processes of a heat engine. Accordingly, a finite time thermodynamic analysis of a solar driven heat engine is carried out considering the irreversibilities related to radiative and convective type of heat transfer modes. The irreversibilities due to these finite rate heat transfers are considered in determining the limits of efficiency and power generation. It is shown that the efficiency of cyclic heat engine operating at maximum power conditions varies between the Carnot and the Curzon–Ahlborn efficiency limits. The variation of the efficiency for the cases applicable to solar
126
driven power engines is discussed through relevant system parameters. Acknowledgements
The author acknowledges the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for this work. REFERENCES [1] Curzon F.L., Ahlborn B., Efficiency of a Carnot engine at maximum power output, Amer. J. Phys. 43 (1975) 22–24. [2] De Vos A., Efficiency of some heat engines at maximum-power conditions, Amer. J. Phys. 53 (1985) 570– 573. [3] Wu C., Power optimization of an endoreversible Brayton gas heat engine, Energy Conversion and Management 31 (6) (1991) 561–565. [4] Chen J., Optimization of a solar-driven heat engine, J. Appl. Phys. 72 (8) (1992) 3778–3780. [5] Chen J., The maximum power output and maximum efficiency of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 27 (1994) 1144–1149. [6] Sahin B., Kodal A., Yavuz H., Efficiency of a Joule– Brayton engine at maximum power density, J. Phys. D: Appl. Phys. 28 (1995) 1309–1313. [7] Sahin B., Kodal A., Yilmaz T., Yavuz H., Maximum power density analysis of an irreversible Joule–Brayton engine, J. Phys. D: Appl. Phys. 29 (1996) 1162–1167. [8] Medina A., Roco J.M.M., Hernandez A.C., Regenerative gas turbines at maximum power density conditions, J. Phys. D: Appl. Phys. 29 (1996) 2802–2805. [9] Cheng C.Y., Chen C.K., Ecological optimization of an endoreversible Brayton cycle, Energy Conversion and Management 39 (1/2) (1998) 33–44. [10] Goktun S., Design parameters of a solar-driven heat engine, Energy Sources 18 (1996) 37–42. [11] Goktun S., Optimization of a solar-driven irreversible Carnot heat engine at maximum power output, Energy Sources 19 (1997) 661–664. [12] Erbay L.B., Yavuz H., An analysis of an endoreversible heat engine with combined heat transfer, J. Phys. D: Appl. Phys. 30 (1997) 2841–2847. [13] Lund K.O., Applications of finite-time thermodynamics to solar power conversion, in: Sieniutycz S., Salamon P. (Eds.), Finite-Time Thermodynamics and Thermoeconomics, Taylor and Francis, New York, 1990. [14] Sahin A.Z., Optimum operating conditions of solar driven heat engines, Energy Conversion and Management 41 (2000) 1335–1343.
Finite-time thermodynamic analysis of a solar driven heat engine Ahmet Z. Sahin Department of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia
(Received 9 December 1999, accepted 13 September 2000)
Abstract — The collective role of radiation and convection modes of heat transfer in a solar driven heat engine is investigated through a finite time thermodynamics analysis. Heat transfer from hot reservoir is assumed to be radiation and/or convection dominated. The irreversibilities due to these finite rate heat transfers were considered in determining the limits of efficiency and power generation that were discussed through varying process parameters. Results were compared with Curzon–Ahlborn and Carnot analysis cases. It is found that the upper limit of efficiency is a function of both the functional temperature dependence of heat transfer and relevant system parameters. 2001 Éditions scientifiques et médicales Elsevier SAS
1. INTRODUCTION
Nomenclature C Q rC rR rT T W
heat transfer coefficient rate of heat transfer . . = CHC /CLC = CHR TH3 /CLC = TL /TH temperature . . . . . . . rate of work . . . . . .
. . . . . . . W·K−1 or W·K−4 . . . . . . . W
. . . . . . . . . . . . . .
Greek symbols η θ
efficiency = Ta /TH
Subscripts a b C CA H HC HR L LC th
boiler side condenser side Carnot Curzon–Ahlborn high temperature reservoir high temperature side convection high temperature side radiation low temperature reservoir low temperature side convection thermal
E-mail address: [email protected] (A.Z. Sahin).
122
K W
The study of irreversible thermodynamic cycles has been undertaken by many researchers after Curzon and Ahlborn’s [1] work. Efficiency of some heat engines where the power output is limited by the rate of heat supply was studied by De Vos [2] at maximum-power conditions. Wu [3] derived a mathematical expression for the power output of an endo-reversible heat engine. Chen [4] studied the optimum operating temperatures in a solar driven heat engine and the optimum performance of the system. Chen [5], later, introduced a new general cyclic model including heat leaks and dissipative process inside the working fluid to optimize the performance of an irreversible heat engine. Sahin et al. [6] studied the efficiency of a Joule–Brayton engine at maximum power density with a consideration of engine size. Their results show that the efficiency at maximum power density is always greater than that presented by Curzon and Ahlborn [1]. In a later publication, they extended their work [7] to include irreversibilities at the compressor and the turbine of a Joule–Brayton engine along with considerations of pressure losses at the burner for the performance analysis. Medina et al. [8] extended the work of Sahin et al. [6, 7] to a regenerative Joule–Brayton cycle where the optimal operating conditions at the engine were expressed in terms of the compressor and turbine isentropic efficiencies and of the heat exchanger effi 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1164-0235(01)00018-8/FLA
A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
ciency. Cheng and Chen [9] studied the effect of regeneration on power output and thermal efficiency of an endoreversible Brayton cycle. Goktun [10] presented a relation between the design parameter of an internally and externally irreversible Carnotlike solar driven heat engine to find the maximum power and the efficiency at maximum power output. He [11] also investigated the optimum performance of an irreversible heat engine system driven by a corrugated sheet solar collector. Erbay and Yavuz [12] analyzed the endo-reversible heat engine by introducing radiation heat transfer coefficients as a function of temperature with combined heat transfer to obtain the design parameters for the operating under the maximum power conditions. The principle of operation of a solar-thermal power plant is presented by Lund [13] in terms of finite heat transfer rates and an internally reversible heat engine by presenting some parametric equations. Finite time thermodynamics analysis of heat engines is usually restricted to systems having either linear of power law heat transfer dependence to the temperatures of both reservoirs and engine working fluid. However, radiation and convection modes of heat transfer are often coupled and play a collective role in processes of heat engine. In a recent study, Sahin [14] investigated the optimum operating conditions of solar driven heat engines with radiation dominated heat transfer from the high temperature reservoir and convection dominated low temperature reservoir heat transfer. As an extension of Sahin’s work [14], both convection and radiation modes of heat transfer are considered from the high temperature reservoir.
2. ANALYSIS Solar driven heat engines used in the atmosphere operate on the simultaneous convection and radiation modes of heat transfer from the high temperature reservoir. Depending on the environmental and operating conditions either of the modes may be dominant. However, in general, the finite time heat transfer amount from high temperature reservoir can be written as QH = CHR TH4 − Ta4 + CHC (TH − Ta ) (1) On the other hand, convection heat transfer is assumed to be the main mode of heat transfer to low temperature reservoir QL = CLC (Tb − TL )
(2)
Figure 1. Solar driven endo-reversible heat engine.
The work output according to the first law is W = QH − QL = ηth QH
(3)
where η = 1 − Tb /Ta . According to the second law of thermodynamics for the endo-reversible heat engine shown in figure 1 QH QL = Ta Tb
(4)
Using equations (1) and (2) in equation (4), a relationship between Tb and Ta is obtained as CHR TH4 − Ta4 CHC TH − Ta −1 Tb = TL 1 − − (5) CLC Ta CLC Ta The optimum temperatures, Ta and Tb , at the maximum power output condition can be obtained by differentiating equation (3) with respect to Ta and setting it equal to zero. After a lengthy algebra the resultant equation to be solved for Ta is found to be 4rR3 θ 11 + rR2 (9rC + 8)θ 8 − 8rR2 (rC + rR )θ 7 + 2rR 3rC2 + 5rC + 2 θ 5 − rR 2(4 + 5rC )(rC + rR ) + 3rT θ 4 + 4rR (rC + rR )2 θ 3 + rC (1 + rC )2 θ 2 − 2rC (1 + rC )(rC + rR )θ + (rC + rR ) rC (rC + rR ) − rT = 0
(6)
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A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
where θ = Ta /TH , rR = (CHR /CLC )TH3 , rC = CHC /CLC , and rT = TL /TH . The solution of equation (6) can be done numerically. It should be noted that the physically meaningful root of θ in equation (6) is the one located between 0 and 1.
3. DISCUSSION From the solution of equation (6), the effect of the radiative (rR ) and convective (rC ) mode of heat transfers on the optimum temperatures (Ta and Tb ) and on the efficiency at maximum power output (η) can be studied, respectively. Figure 2(a) shows the variation of θ with respect to rR for three different values of rT and rC = 0.1. As radiation (rR ) increases θ increases and approaches to 1. This means that the optimal temperature Ta gets closer to the source temperature TH as radiation increases. For higher values of rT , the ratio of the optimum temperature Ta to the source temperature TH is higher in general. For no radiation case, as rR → 0 (convection mode only), it can be easily found from equation (6) that √ rC + rT θ→ (7) 1 + rC
(a)
This indicates that as rT (i.e., TL ) increases θ (i.e., the optimum value of Ta for fixed value of TH ) increases. Since Ta /TL = θ/rT , using equation (7) √ rC + rT Ta = TL rT (1 + rC ) The optimum value of Tb , on the other hand, becomes √ rC + rT Tb =√ TL rT (1 + rC ) Thus the efficiency for this case approaches the Curzon– Ahlborn efficiency √ η → ηCA = 1 − rT As radiation dominates rR → ∞, it can be shown that, from equation (6), θ 8 − 2θ 4 + 1 = 0 or
124
4 2 θ −1 =0
(b) Figure 2. Solution of θ in equation (6) for three different cases of rT : (a) θ versus rR for rC = 0.1, and (b) θ versus rC for rR = 0.1.
The only meaningful root of this equation is +1, i.e., θ =1 This means that Ta /TL = 1/rT and Tb /TL = 1. Thus the efficiency in this case approaches the Carnot efficiency η → ηC = 1 − rT Figure 2(b) shows the effect of convection (rC ) on the optimum temperature variation θ for fixed value of rT = 0.1 and three different values of rT . θ reaches a
A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
(a)
(a)
(b) (b) Figure 3. Optimum temperature variations of Ta and Tb for the case rT = 0.1: (a) Temperature variations with respect to rR for rC = 0.1; (b) Temperature variations with respect to rC for rR = 0.1.
weak minimum and then increases as rC is increased. Higher values of rT yields higher values of θ . Variation of optimum temperatures, Ta and Tb , versus rR is shown in figure 3(a) where both rC and rT are fixed to be equal to 0.1. As rR approaches 1, the difference between Tb and TL increases. Tb approaches to a constant value as rR increases. The optimum temperature variations with respect to rC are given in figure 3(b) for fixed values of rR = 0.1
Figure 4. Variation of efficiency (η ) for constant value of rT = 0.1 and three cases of rC : (a) Efficiency variation with respect to rR ; (b) Efficiency variation with respect to rC .
and rT = 0.1. Tb /TL increases while Ta /TL initially decreases to a minimum and then increases as rC increases. The gap between Ta and Tb also increases. The variation of efficiency at maximum power output (η) for the present case is given with respect to rR in figure 4(a) for a fixed value of rT = 0.1 and three cases of rC . Since rT is taken to be constant, both Carnot and Curzon–Ahlborn efficiencies, (ηC ) and (ηCA ), are constant and equal to 0.9 and 0.6838, respectively. ηCA is shown in a dotted line in figure 4(a). It is clear that η is higher than ηCA for the whole range of rR values. The difference in these efficiencies is larger for smaller
125
A.Z. Sahin / Exergy Int. J. 1(2) (2001) 122–126
rC values. As rR is increased η reaches to a maximum and then decreases. Finally, the effect of convection on the efficiency is given in figure 4(b) for fixed value of rT = 0.1 and three different values of rR . Curzon–Ahlborn efficiency is also included in figure 4(b) in a dotted line for reference. Clearly the efficiency may become much higher than the Curzon–Ahlborn efficiency, especially for low values of rC for which case the radiation becomes the dominant mode of heat transfer from the source temperature TH . As convection (rC ) increases, however, the efficiency approaches to Curzon–Ahlborn efficiency for all values of rR . As a numerical example, consider the case where rT = 0.1, rC = 0.1 and rR = 0.2, the efficiency becomes nearly 73% which is 4.5% higher than the Curzon–Ahlborn efficiency. As rR increases to 0.8 the efficiency decreases to 69.5%. This indicated a decrease of 3.5% as compared to the case of rR = 0.2. On the other hand, when rR decreases to 0 the efficiency increases sharply as shown in figure 4(b). As rC is increased, however, the effect of rR on the efficiency diminishes as seen in figure 4 (a) and (b). As a result, both convection and radiation modes of heat transfer from the high temperature reservoir need to be considered to assess the efficiency accurately, since variation in the either of the modes may influence the efficiency considerably.
4. CONCLUSIONS Radiation and convection modes of heat transfer are often coupled and play a collective role in processes of a heat engine. Accordingly, a finite time thermodynamic analysis of a solar driven heat engine is carried out considering the irreversibilities related to radiative and convective type of heat transfer modes. The irreversibilities due to these finite rate heat transfers are considered in determining the limits of efficiency and power generation. It is shown that the efficiency of cyclic heat engine operating at maximum power conditions varies between the Carnot and the Curzon–Ahlborn efficiency limits. The variation of the efficiency for the cases applicable to solar
126
driven power engines is discussed through relevant system parameters. Acknowledgements
The author acknowledges the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for this work. REFERENCES [1] Curzon F.L., Ahlborn B., Efficiency of a Carnot engine at maximum power output, Amer. J. Phys. 43 (1975) 22–24. [2] De Vos A., Efficiency of some heat engines at maximum-power conditions, Amer. J. Phys. 53 (1985) 570– 573. [3] Wu C., Power optimization of an endoreversible Brayton gas heat engine, Energy Conversion and Management 31 (6) (1991) 561–565. [4] Chen J., Optimization of a solar-driven heat engine, J. Appl. Phys. 72 (8) (1992) 3778–3780. [5] Chen J., The maximum power output and maximum efficiency of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 27 (1994) 1144–1149. [6] Sahin B., Kodal A., Yavuz H., Efficiency of a Joule– Brayton engine at maximum power density, J. Phys. D: Appl. Phys. 28 (1995) 1309–1313. [7] Sahin B., Kodal A., Yilmaz T., Yavuz H., Maximum power density analysis of an irreversible Joule–Brayton engine, J. Phys. D: Appl. Phys. 29 (1996) 1162–1167. [8] Medina A., Roco J.M.M., Hernandez A.C., Regenerative gas turbines at maximum power density conditions, J. Phys. D: Appl. Phys. 29 (1996) 2802–2805. [9] Cheng C.Y., Chen C.K., Ecological optimization of an endoreversible Brayton cycle, Energy Conversion and Management 39 (1/2) (1998) 33–44. [10] Goktun S., Design parameters of a solar-driven heat engine, Energy Sources 18 (1996) 37–42. [11] Goktun S., Optimization of a solar-driven irreversible Carnot heat engine at maximum power output, Energy Sources 19 (1997) 661–664. [12] Erbay L.B., Yavuz H., An analysis of an endoreversible heat engine with combined heat transfer, J. Phys. D: Appl. Phys. 30 (1997) 2841–2847. [13] Lund K.O., Applications of finite-time thermodynamics to solar power conversion, in: Sieniutycz S., Salamon P. (Eds.), Finite-Time Thermodynamics and Thermoeconomics, Taylor and Francis, New York, 1990. [14] Sahin A.Z., Optimum operating conditions of solar driven heat engines, Energy Conversion and Management 41 (2000) 1335–1343.