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Heating load vs. COP characteristic of an endoreversible Carnot heat-pump subjected to the heat-transfer law q ∝ (ΔTn)m
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APPLIED ENERGY Applied Energy 85 (2008) 96–100 www.elsevier.com/locate/apenergy
Heating load vs. COP characteristic of an endoreversible Carnot heat-pump subjected to the heat-transfer law q / (DT n)m Jun Li, Lingen Chen *, Fengrui Sun Postgraduate School, Naval University of Engineering, Wuhan 430033, PR China Available online 4 September 2007
Abstract The relation between heating load and coefficient of performance (COP) of an endoreversible Carnot heat-pump is derived based on a new generalized convective heat-transfer law and generalized radiative heat-transfer law, q / (DT n)m. Our results include those obtained in many literature studies and can provide some theoretical guidance for the designs of real heat pumps. 2007 Elsevier Ltd. All rights reserved. Keywords: Finite-time thermodynamics; Endoreversible Carnot heat-pump; Optimal performance; Heat-transfer law
1. Introduction Since the 1970s, the research into identifying the performance limits of thermodynamic processes and optimization of thermodynamic cycles has made tremendous progress by using finite-time thermodynamics [1–8]. Blanchard [9] was the first to extend the Curzon–Ahlborn analysis method [10] to the analysis of heat-pump cycles, and derived the coefficient of performance (COP) bounds for the fixed heating load for a Newtown’s law endoreversible Carnot heat-pump. Goth and Feidt [11], Feidt [12], Philippi and Feidt [13] and Feidt [14] derived the optimal COP for the fixed heating-load, i.e. the fundamental optimal relation of a Newtown’s law Carnot heat-pump. Sun et al. [15,16] extended the characteristic parameters of heat engines to the heat pump and derived the optimization criteria of a steady-flow two-heat-reservoir heat-pump. Wu [17], Chen et al. [18] and Wu et al. [19] investigated the specific heating-load optimization of the endoreversible Carnot heat-pump, derived the bounds of specific heating-load and COP as well as the optimal relation between the optimal specific heating-load and COP. In general, heat transfer between the working fluid and the heat reservoirs is not necessarily Newtonian. Some authors have assessed the effect of the heat-transfer laws on the performance of endoreversible Carnot heat-pump [20–25]. Chen et al. [20] first derived the optimal relation between heating load and COP of a *
Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.013
J. Li et al. / Applied Energy 85 (2008) 96–100
97
Carnot heat-pump with the linear phenomenological heat-transfer law q / D (T 1). Sun et al. [21] first obtained the performance limits and the optimal relation between heating load and COP of the endoreversible Carnot heat-pump with the generalized radiative heat-transfer law q / (DT n). The other optimal performances of an endoreversible and irreversible Carnot heat-pump were obtained based on this heat-transfer law [22,25]. Chen et al. [23] first derived the optimal relation between heating load and COP of a Carnot heat pump with the generalized convective heat-transfer law q / (DT)n. The other optimal performances of an endoreversible and irreversible Carnot heat-pump were obtained based on this heat-transfer law [24,25]. One of the aims of finite-time thermodynamics is to pursue generalized rules and results. This paper will extend the previous work by using a new generalized heat-transfer law, including a generalized convective heat-transfer law and a generalized radiative heat-transfer law, q / (DT n)m in the heat-transfer processes between the heat pump and its surroundings to find the optimal relationship for the endoreversible Carnot heat-pump.
2. Endoreversible Carnot heat-pump model An endoreversible Carnot heat-pump and its surroundings to be considered in this paper is shown in Fig. 1. The following assumptions are made for this model: • The working fluid (refrigerant) flows through the system in a steady-state fashion. The cycle consists of two isothermal and two adiabatic processes. • Because of the heat resistance, the working-fluid’s temperatures (THC and TLC) are different from the reservoir’s temperatures (TH and TL). The four temperatures are of the following decreasing order: THC > TH > TL > TLC. The heat-transfer law is q / (DT n)m. The heat-transfer surface areas (F1 and F2) of the high- and low-temperature heat-exchangers are finite. The total heat transfer surface area (F) of the two heat exchangers is assumed to be a constant: F = F1 + F2. • The rate of heat transfer released to the heat sink is QH and the rate of heat transfer supplied by the heat source is QL. QH is the heating load of the heat pump. • The heat pump is an endoreversible one, i.e. the only irreversibility is due to the finite-rate heat-transfer.
TH QH
α F1
T HC
Endoreversible Carnot heatpump
P
TLC
QL
β F2
TL Fig. 1. Endoreversible Carnot heat-pump model.
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J. Li et al. / Applied Energy 85 (2008) 96–100
3. Generalized optimal characteristics The second law of thermodynamics requires that QH =T HC ¼ QL =T LC
ð1Þ
The first law of thermodynamics gives the heating load (p) and COP (u) of the heat pump as p ¼ QH
ð2Þ
u ¼ QH =P ¼ QH =ðQH QL Þ
ð3Þ
and respectively, where P is the power input to the heat pump. Consider that the heat transfer between the heat pump and its surroundings follows a new generalized convective heat-transfer law and a generalized radiative heat-transfer law. Then m
p ¼ QH ¼ aF 1 ðT nHC T nH Þ and
QL ¼ bF 2 ðT nL T nLC Þ
m
ð4Þ
respectively, where a is the overall heat-transfer coefficient of the high-temperature-side heat-exchanger and b is the overall heat-transfer coefficient of the low-temperature-side heat exchanger. Defining the heat-transfer surface area ratio (f) and working-fluid temperature ratio (x) as follows: f = F1/F2 and x = TLC THC, where 0 6 x 6 TL/TH. From Eqs. (1)–(4), one can obtain u ¼ 1=ð1 xÞ " #m afF T nL T nH xn p¼ 1 þ f ðrfxÞ1=m þ xn
ð5Þ ð6Þ
where r = a/b. Eq. (6) indicates that the heating load of the endoreversible Carnot heat-pump is a function of the heat-transfer surface area ratio (f) for given TH, TL, a, b, n, m and x. Taking the derivative of p with respect to f and setting it equal to zero (dp/df = 0) yields fa ¼ ðxnm1 =rÞ1=ðmþ1Þ
ð7Þ
The corresponding optimal heating load is p¼
aF ðT nL xn T nH Þ
m
ð8Þ
1=ðmþ1Þ mþ1
½1 þ ðrx1mn Þ
From Eqs. (5) and (8), one can obtain the optimal relation between the heating load p and the COP u, namely p¼
aF ½T nL un =ðu 1Þn T nH m f1 þ ½rð1 1=uÞ
1mn 1=ðmþ1Þ mþ1
g
ð9Þ
Eq. (9) is the fundamental optimal relation between the heating load and the COP of the endoreversible Carnot heat-pump with the heat transfer law of q / (DTn)m. One can see that the heating load is a monotonicallydecreasing function of the COP. 4. Discussion When m = 1, Eq. (9) becomes p¼
aF ½T nL un =ðu 1Þn T nH f1 þ ½rð1 1=uÞ
1n 1=2 2
g
ð10Þ
It is the same result as that obtained in Refs. [21,25]. Eq. (10) indicates that p is a monotonically-decreasing function of u. If n = 1, it is the result of the heat pump with Newtown heat-transfer law [9,11–19]. If n = 1, it is the result of the heat pump experiencing a linear phenomenological heat-transfer law [20,21]. If n = 4, it is the result of the heat pump with the radiative heat-transfer law [21,25].
J. Li et al. / Applied Energy 85 (2008) 96–100
99
Fig. 2. Heating load versus COP relationship for m = 1.25 and n = 4.
When n = 1, Eq. (9) becomes m
p¼
aF ½T L u=ðu 1Þ T H f1 þ ½rð1 1=uÞ
1m 1=ðmþ1Þ mþ1
g
ð11Þ
This is the same result as that obtained in Refs. [23,25]. If m = 1, it is the result of the heat pump experiencing the Newtown heat-transfer law [9,11–19]. If m = 1.25, it is the result of the heat pump [23,25] under the Dulong–Petit heat transfer law [26]. 5. Numerical example To show the heating load vs. COP characteristic of the Carnot heat-pump with the generalized heat-transfer law, one numerical example is provided, see Fig. 2. In the numerical calculation for the performance characteristics of the endoreversible Carnot heat pump, TL = 273 K, TH = 300 K, aF = 4 W/K, a = b(r = 1), n = 4 and m = 1.25 are set. It shows that the relation between heating load and COP of the heat pump is a monotonically-decreasing curve. 6. Conclusion The fundamental optimal relation between the heating load and the COP of an endoreversible Carnot heat pump is derived by using a new generalized heat-transfer law, including a generalized convective heat-transfer law and a generalized radiative heat-transfer law, q / D (Tn)m. Our results include those obtained in the many literature and can provide some theoretical guidance for the designs of practical heat-pumps. Acknowledgements This paper is supported by the Program for New Century Excellent Talents in the Universities of PR China (Project No. 20041006) and The Foundation for the Authors of Nationally Excellent Doctoral Dissertations of PR China (Project No. 200136). References [1] Andresen B. Finite-time thermodynamics. Physics Laboratory II. University of Copenhagen; 1983. [2] De Vos A. Endoreversible thermodynamics of solar energy conversion. Oxford: Oxford University Press; 1992. [3] Bejan A. Entropy-generation minimization: the new thermodynamics of finite-size devices and finite-time processes. J Appl Phys 1996;79(3):1191–218.
100
J. Li et al. / Applied Energy 85 (2008) 96–100
[4] Feidt M. Thermodynamique et optimisation energetique des systems et procedes. Second ed. Paris: Technique et Documentation, Lavoisier; 1996. [5] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy-generation minimization of energy systems. J NonEquilib Thermodynam 1999;24(4):327–59. [6] Berry RS, Kazakov VA, Sieniutycz S, et al.. Thermodynamic optimization of finite-time processes. Chichester: Wiley; 1999. [7] Chen L, Sun F. Advances in finite-time thermodynamics: analysis and optimization. New York: Nova Science Publishers; 2004. [8] Chen L. Finite-time thermodynamic analysis of irreversible processes and cycles. Beijing: Higher Education Press; 2005, in Chinese. [9] Blanchard CH. Coefficient of performance for finite-speed heat pump. J Appl Phys 1980;51(5):2471–2. [10] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power-output. Am J Phys 1975;43(1):22–4. [11] Goth Y, Feidt M. Optimum COP for endoreversible heat-pump or refrigerating machine. CR Acad Sci, Paris 1986;303(1):19–24. [12] Feidt M. Finite-time thermodynamics applied to optimization of heat pumps and refrigerating machine cycles. In: 12th IMACS world congress on scientific computation, Paris, France, 1988. [13] Philippi I, Feidt M. Finite-time thermodynamics applied to inverse cycle machines. In: XVIII International congress on refrigeration, Montreal, Canada, 1991. [14] Feidt M. Sur une systematique des cycles imparfaits. Entropie 1997;205(1):53–61. [15] Sun F, Chen W, Chen L. An analysis of the holographic spectrum and selection of the optimum parameters for a reverse endoreversible Carnot cycle. J Naval Acad Eng 1990(2):40–5, in Chinese. [16] Chen W, Sun F, Chen L. Finite-time thermodynamic criteria for selecting parameters of refrigerating and heat-pumping cycles between heat reservoirs. Chinese Sci Bull 1990;35(19):1670–2. [17] Wu C. Specific heating load of an endoreversible Carnot heat-pump. Int J Ambient Energ 1993;14(1):25–8. [18] Chen L, Wu C, Sun F. Heat-transfer effect on the specific heating load of heat pumps. Appl Therm Eng 1997;17(1):103–10. [19] Wu C, Chen L, Sun F. Optimization of steady flow heat-pumps. Energ Convers Manage 1998;39(5/6):445–53. [20] Chen L, Sun F, Chen W. The relationship between COP and heating load of a Carnot heat-pump. J Eng Therm Energ Power 1990;5(3):48–52, in Chinese. [21] Sun F, Chen W, Chen L, Wu C. Optimal performance of an endoreversible Carnot heat pump. Energ Convers Manage 1997;38(14):1439–43. [22] Ni N, Chen L, Sun F, Wu C. Effect of heat-transfer law on the performance of a generalized irreversible Carnot heat-pump. J Inst Energ 1999;72(491):64–8. [23] Chen W, Sun F, Cheng S, Chen L. Study on optimal performance and working temperature of endoreversible forward and reverse Carnot-cycles. Int J Energ Res 1995;19(9):751–9. [24] Zhu X, Chen L, Sun F, Wu C. Optimal performance of a generalized irreversible Carnot heat-pump with a generalized heat-transfer law. Phys Scripta 2001;64(6):584–7. [25] Feidt M. Thermodynamics and optimization of reverse-cycle machines. In: Bejan A, Mamut E, editors. Thermodynamic optimization of complex energy-systems. Dordrecht: Kluwer Academic Press; 1999. p. 385–401. [26] O’Sullivan CT. Newton’s law of cooling – a critical assessment. Am J Phys 1990;58(12):956–60.
APPLIED ENERGY Applied Energy 85 (2008) 96–100 www.elsevier.com/locate/apenergy
Heating load vs. COP characteristic of an endoreversible Carnot heat-pump subjected to the heat-transfer law q / (DT n)m Jun Li, Lingen Chen *, Fengrui Sun Postgraduate School, Naval University of Engineering, Wuhan 430033, PR China Available online 4 September 2007
Abstract The relation between heating load and coefficient of performance (COP) of an endoreversible Carnot heat-pump is derived based on a new generalized convective heat-transfer law and generalized radiative heat-transfer law, q / (DT n)m. Our results include those obtained in many literature studies and can provide some theoretical guidance for the designs of real heat pumps. 2007 Elsevier Ltd. All rights reserved. Keywords: Finite-time thermodynamics; Endoreversible Carnot heat-pump; Optimal performance; Heat-transfer law
1. Introduction Since the 1970s, the research into identifying the performance limits of thermodynamic processes and optimization of thermodynamic cycles has made tremendous progress by using finite-time thermodynamics [1–8]. Blanchard [9] was the first to extend the Curzon–Ahlborn analysis method [10] to the analysis of heat-pump cycles, and derived the coefficient of performance (COP) bounds for the fixed heating load for a Newtown’s law endoreversible Carnot heat-pump. Goth and Feidt [11], Feidt [12], Philippi and Feidt [13] and Feidt [14] derived the optimal COP for the fixed heating-load, i.e. the fundamental optimal relation of a Newtown’s law Carnot heat-pump. Sun et al. [15,16] extended the characteristic parameters of heat engines to the heat pump and derived the optimization criteria of a steady-flow two-heat-reservoir heat-pump. Wu [17], Chen et al. [18] and Wu et al. [19] investigated the specific heating-load optimization of the endoreversible Carnot heat-pump, derived the bounds of specific heating-load and COP as well as the optimal relation between the optimal specific heating-load and COP. In general, heat transfer between the working fluid and the heat reservoirs is not necessarily Newtonian. Some authors have assessed the effect of the heat-transfer laws on the performance of endoreversible Carnot heat-pump [20–25]. Chen et al. [20] first derived the optimal relation between heating load and COP of a *
Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.013
J. Li et al. / Applied Energy 85 (2008) 96–100
97
Carnot heat-pump with the linear phenomenological heat-transfer law q / D (T 1). Sun et al. [21] first obtained the performance limits and the optimal relation between heating load and COP of the endoreversible Carnot heat-pump with the generalized radiative heat-transfer law q / (DT n). The other optimal performances of an endoreversible and irreversible Carnot heat-pump were obtained based on this heat-transfer law [22,25]. Chen et al. [23] first derived the optimal relation between heating load and COP of a Carnot heat pump with the generalized convective heat-transfer law q / (DT)n. The other optimal performances of an endoreversible and irreversible Carnot heat-pump were obtained based on this heat-transfer law [24,25]. One of the aims of finite-time thermodynamics is to pursue generalized rules and results. This paper will extend the previous work by using a new generalized heat-transfer law, including a generalized convective heat-transfer law and a generalized radiative heat-transfer law, q / (DT n)m in the heat-transfer processes between the heat pump and its surroundings to find the optimal relationship for the endoreversible Carnot heat-pump.
2. Endoreversible Carnot heat-pump model An endoreversible Carnot heat-pump and its surroundings to be considered in this paper is shown in Fig. 1. The following assumptions are made for this model: • The working fluid (refrigerant) flows through the system in a steady-state fashion. The cycle consists of two isothermal and two adiabatic processes. • Because of the heat resistance, the working-fluid’s temperatures (THC and TLC) are different from the reservoir’s temperatures (TH and TL). The four temperatures are of the following decreasing order: THC > TH > TL > TLC. The heat-transfer law is q / (DT n)m. The heat-transfer surface areas (F1 and F2) of the high- and low-temperature heat-exchangers are finite. The total heat transfer surface area (F) of the two heat exchangers is assumed to be a constant: F = F1 + F2. • The rate of heat transfer released to the heat sink is QH and the rate of heat transfer supplied by the heat source is QL. QH is the heating load of the heat pump. • The heat pump is an endoreversible one, i.e. the only irreversibility is due to the finite-rate heat-transfer.
TH QH
α F1
T HC
Endoreversible Carnot heatpump
P
TLC
QL
β F2
TL Fig. 1. Endoreversible Carnot heat-pump model.
98
J. Li et al. / Applied Energy 85 (2008) 96–100
3. Generalized optimal characteristics The second law of thermodynamics requires that QH =T HC ¼ QL =T LC
ð1Þ
The first law of thermodynamics gives the heating load (p) and COP (u) of the heat pump as p ¼ QH
ð2Þ
u ¼ QH =P ¼ QH =ðQH QL Þ
ð3Þ
and respectively, where P is the power input to the heat pump. Consider that the heat transfer between the heat pump and its surroundings follows a new generalized convective heat-transfer law and a generalized radiative heat-transfer law. Then m
p ¼ QH ¼ aF 1 ðT nHC T nH Þ and
QL ¼ bF 2 ðT nL T nLC Þ
m
ð4Þ
respectively, where a is the overall heat-transfer coefficient of the high-temperature-side heat-exchanger and b is the overall heat-transfer coefficient of the low-temperature-side heat exchanger. Defining the heat-transfer surface area ratio (f) and working-fluid temperature ratio (x) as follows: f = F1/F2 and x = TLC THC, where 0 6 x 6 TL/TH. From Eqs. (1)–(4), one can obtain u ¼ 1=ð1 xÞ " #m afF T nL T nH xn p¼ 1 þ f ðrfxÞ1=m þ xn
ð5Þ ð6Þ
where r = a/b. Eq. (6) indicates that the heating load of the endoreversible Carnot heat-pump is a function of the heat-transfer surface area ratio (f) for given TH, TL, a, b, n, m and x. Taking the derivative of p with respect to f and setting it equal to zero (dp/df = 0) yields fa ¼ ðxnm1 =rÞ1=ðmþ1Þ
ð7Þ
The corresponding optimal heating load is p¼
aF ðT nL xn T nH Þ
m
ð8Þ
1=ðmþ1Þ mþ1
½1 þ ðrx1mn Þ
From Eqs. (5) and (8), one can obtain the optimal relation between the heating load p and the COP u, namely p¼
aF ½T nL un =ðu 1Þn T nH m f1 þ ½rð1 1=uÞ
1mn 1=ðmþ1Þ mþ1
g
ð9Þ
Eq. (9) is the fundamental optimal relation between the heating load and the COP of the endoreversible Carnot heat-pump with the heat transfer law of q / (DTn)m. One can see that the heating load is a monotonicallydecreasing function of the COP. 4. Discussion When m = 1, Eq. (9) becomes p¼
aF ½T nL un =ðu 1Þn T nH f1 þ ½rð1 1=uÞ
1n 1=2 2
g
ð10Þ
It is the same result as that obtained in Refs. [21,25]. Eq. (10) indicates that p is a monotonically-decreasing function of u. If n = 1, it is the result of the heat pump with Newtown heat-transfer law [9,11–19]. If n = 1, it is the result of the heat pump experiencing a linear phenomenological heat-transfer law [20,21]. If n = 4, it is the result of the heat pump with the radiative heat-transfer law [21,25].
J. Li et al. / Applied Energy 85 (2008) 96–100
99
Fig. 2. Heating load versus COP relationship for m = 1.25 and n = 4.
When n = 1, Eq. (9) becomes m
p¼
aF ½T L u=ðu 1Þ T H f1 þ ½rð1 1=uÞ
1m 1=ðmþ1Þ mþ1
g
ð11Þ
This is the same result as that obtained in Refs. [23,25]. If m = 1, it is the result of the heat pump experiencing the Newtown heat-transfer law [9,11–19]. If m = 1.25, it is the result of the heat pump [23,25] under the Dulong–Petit heat transfer law [26]. 5. Numerical example To show the heating load vs. COP characteristic of the Carnot heat-pump with the generalized heat-transfer law, one numerical example is provided, see Fig. 2. In the numerical calculation for the performance characteristics of the endoreversible Carnot heat pump, TL = 273 K, TH = 300 K, aF = 4 W/K, a = b(r = 1), n = 4 and m = 1.25 are set. It shows that the relation between heating load and COP of the heat pump is a monotonically-decreasing curve. 6. Conclusion The fundamental optimal relation between the heating load and the COP of an endoreversible Carnot heat pump is derived by using a new generalized heat-transfer law, including a generalized convective heat-transfer law and a generalized radiative heat-transfer law, q / D (Tn)m. Our results include those obtained in the many literature and can provide some theoretical guidance for the designs of practical heat-pumps. Acknowledgements This paper is supported by the Program for New Century Excellent Talents in the Universities of PR China (Project No. 20041006) and The Foundation for the Authors of Nationally Excellent Doctoral Dissertations of PR China (Project No. 200136). References [1] Andresen B. Finite-time thermodynamics. Physics Laboratory II. University of Copenhagen; 1983. [2] De Vos A. Endoreversible thermodynamics of solar energy conversion. Oxford: Oxford University Press; 1992. [3] Bejan A. Entropy-generation minimization: the new thermodynamics of finite-size devices and finite-time processes. J Appl Phys 1996;79(3):1191–218.
100
J. Li et al. / Applied Energy 85 (2008) 96–100
[4] Feidt M. Thermodynamique et optimisation energetique des systems et procedes. Second ed. Paris: Technique et Documentation, Lavoisier; 1996. [5] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy-generation minimization of energy systems. J NonEquilib Thermodynam 1999;24(4):327–59. [6] Berry RS, Kazakov VA, Sieniutycz S, et al.. Thermodynamic optimization of finite-time processes. Chichester: Wiley; 1999. [7] Chen L, Sun F. Advances in finite-time thermodynamics: analysis and optimization. New York: Nova Science Publishers; 2004. [8] Chen L. Finite-time thermodynamic analysis of irreversible processes and cycles. Beijing: Higher Education Press; 2005, in Chinese. [9] Blanchard CH. Coefficient of performance for finite-speed heat pump. J Appl Phys 1980;51(5):2471–2. [10] Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power-output. Am J Phys 1975;43(1):22–4. [11] Goth Y, Feidt M. Optimum COP for endoreversible heat-pump or refrigerating machine. CR Acad Sci, Paris 1986;303(1):19–24. [12] Feidt M. Finite-time thermodynamics applied to optimization of heat pumps and refrigerating machine cycles. In: 12th IMACS world congress on scientific computation, Paris, France, 1988. [13] Philippi I, Feidt M. Finite-time thermodynamics applied to inverse cycle machines. In: XVIII International congress on refrigeration, Montreal, Canada, 1991. [14] Feidt M. Sur une systematique des cycles imparfaits. Entropie 1997;205(1):53–61. [15] Sun F, Chen W, Chen L. An analysis of the holographic spectrum and selection of the optimum parameters for a reverse endoreversible Carnot cycle. J Naval Acad Eng 1990(2):40–5, in Chinese. [16] Chen W, Sun F, Chen L. Finite-time thermodynamic criteria for selecting parameters of refrigerating and heat-pumping cycles between heat reservoirs. Chinese Sci Bull 1990;35(19):1670–2. [17] Wu C. Specific heating load of an endoreversible Carnot heat-pump. Int J Ambient Energ 1993;14(1):25–8. [18] Chen L, Wu C, Sun F. Heat-transfer effect on the specific heating load of heat pumps. Appl Therm Eng 1997;17(1):103–10. [19] Wu C, Chen L, Sun F. Optimization of steady flow heat-pumps. Energ Convers Manage 1998;39(5/6):445–53. [20] Chen L, Sun F, Chen W. The relationship between COP and heating load of a Carnot heat-pump. J Eng Therm Energ Power 1990;5(3):48–52, in Chinese. [21] Sun F, Chen W, Chen L, Wu C. Optimal performance of an endoreversible Carnot heat pump. Energ Convers Manage 1997;38(14):1439–43. [22] Ni N, Chen L, Sun F, Wu C. Effect of heat-transfer law on the performance of a generalized irreversible Carnot heat-pump. J Inst Energ 1999;72(491):64–8. [23] Chen W, Sun F, Cheng S, Chen L. Study on optimal performance and working temperature of endoreversible forward and reverse Carnot-cycles. Int J Energ Res 1995;19(9):751–9. [24] Zhu X, Chen L, Sun F, Wu C. Optimal performance of a generalized irreversible Carnot heat-pump with a generalized heat-transfer law. Phys Scripta 2001;64(6):584–7. [25] Feidt M. Thermodynamics and optimization of reverse-cycle machines. In: Bejan A, Mamut E, editors. Thermodynamic optimization of complex energy-systems. Dordrecht: Kluwer Academic Press; 1999. p. 385–401. [26] O’Sullivan CT. Newton’s law of cooling – a critical assessment. Am J Phys 1990;58(12):956–60.