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Cooling-load density optimization for a regenerated air refrigerator
Applied Energy 78 (2004) 315–328 www.elsevier.com/locate/apenergy
Cooling-load density optimization for a regenerated air refrigerator Shengbing Zhoua, Lingen Chena,*, Fengrui Suna, Chih Wub a Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA
b
Accepted 21 August 2003
Abstract A performance analysis and optimization of a regenerated air refrigeration cycle with variable-temperature heat-reservoirs is carried out by taking the cooling-load density, i.e., the ratio of cooling load to the maximum specific volume in the cycle, as the optimization objective using finite-time thermodynamics (FTT) or entropy-generation minimization (EGM). The model of a regenerated air refrigerator is presented, and analytical relationships between cooling-load density and pressure ratio, as well as between coefficient of performance (COP) and pressure ratio are derived. The irreversibilities considered in the analysis include the heattransfer losses in the hot- and cold-side heat-exchangers and the regenerator, the non-isentropic compression and expansion losses in the compressor and expander, and the pressuredrop losses in the piping. The cycle performance comparison under maximum cooling-load density and maximum cooling-load conditions is performed via detailed numerical calculations. The optimal performance characteristics of the cycle are obtained by optimizing the pressure ratio of the compressor, and searching for the optimum distribution of heat-conductances of the hot- and cold-side heat-exchangers and regenerator for the fixed total heatexchanger inventory. The effect of heat capacity rate matching between the working fluid and heat reservoirs on the cooling-load density is analyzed for the cycle. The influences of the effectiveness of the regenerator as well as the hot- and cold-side heat-exchangers, the efficiencies of the expander and the compressor, the pressure-recovery coefficient, and the temperature ratio of the heat reservoirs on the cooling-load density and COP are examined and illustrated by numerical examples. # 2003 Elsevier Ltd. All rights reserved. Keywords: Regenerated air refrigeration cycle; Variable-temperature heat reservoirs; Cooling-load density; Optimization
* Corresponding author. Tel.: 0086-27-83615046; fax: 0086-27-83638709. E-mail address: [email protected], [email protected] (L. Chen). 0306-2619/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2003.08.008
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1. Introduction Research upon air refrigeration cycles, in which air is the working fluid, has been paid more attention recently due to the destruction of the Ozonosphere by chlorofluorocarbon (CFC) and the pressure of environmental protection [1–3]. Optimizations of air cycles are also carried out using finite-time thermodynamics (FTT) or entropy-generation minimization (EGM) [4–7]. Chen et al. investigated the cooling-load versus COP characteristics of a simple [8] and a regenerated [9,10] air refrigeration cycle with heat-transfer loss and/or other irreversibilities. Luo et al. [11] optimized the cooling-load and the COP of a simple irreversible air refrigeration cycle by searching for the optimum pressure-ratio of the compressor and the optimum distribution of heat conductance of the hot- and cold-side heat-exchangers for the fixed total heat-exchanger inventory. Some authors, including Erbay and Yavuz [12], and Yavuz and Erbay [13], have taken the cooling-load density, defined as the ratio of cooling-load to the maximum specific volume in the cycle, as the optimization objective and analyzed the optimum performance of the Stirling refrigerator as well as the Ericsson refrigerator with a realistic regenerator due to the polytropic processes applied for the heat absorption and rejection. Zhou et al. analyzed and optimized the cooling-load density of endoreversible simple air refrigeration cycles coupled to constant- [14] and variable[15] temperature heat-reservoirs, and of an irreversible simple air refrigeration cycle coupled to constant-temperature heat-reservoirs [16]. The aim of this paper is to analyze and optimize the performance of a regenerated air refrigeration cycle with variable-temperature heat-reservoirs, which is much closer to the real air refrigerator, using a cooling-load density objective based on the model presented by Chen et al. [14]. The analytical formula about cooling-load density is derived by taking into account heat-resistance losses in the hot- and coldside heat-exchangers, the non-isentropic expansion and compression losses in the expander and compressor, and the pressure-drop loss in the piping. The coolingload density objective is to optimize the cycle performance including the refrigerator size effects.
2. Air refrigerator cycle A regenerated air refrigeration cycle with variable-temperature heat-reservoirs and its surroundings, to be considered in this paper, are shown in Fig. 1(a) and (b). Between 2 and 3 in the TS diagram is a non-isentropic compression process in the compressor. Process 3-6 is heat rejection to the heat sink in the hot-side heat-exchanger. Process 6-4 is heat rejection in the regenerator. Process 4-1 is non- isentropic expansion process in the expander. Process 1-5 is a heat addition (cooling) process from the heat source in the cold-side heat-exchanger. Process 5-2 is the heat addition process in the regenerator. Processes 2-3s and 4-1s are isentropic compression and expansion in the ideal compressor and expander corresponding to the non-isentropic processes 2-3 and 4-1, respectively. The compressor and expansion efficiencies are defined as:
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Fig. 1. (a) The schematic of the regenerated-air refrigerator A—the latter cooler; B—the refrigerator; C—the compressor; R—the regenerator; T—the turbine cooler (expander). (b) The temperature–entropy diagram of the regenerated-air refrigeration cycle.
c ¼ ðT3s T2 Þ=ðT3 T2 Þ; t ¼ ðT4 T1 Þ=ðT4 T1s Þ
ð1Þ
The pressure ratio of the compressor is ¼ P3 =P2 , where P is the pressure. The pressure-drop P in the piping is expressed by the pressure recovery coefficient D ¼ ð1 DP12 =P1 Þð1 DP34 =P4 Þ. The pressure ratio of the expansion process is P4 =P1 ¼ DP3 =P2 ¼ D.
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The high-temperature (hot-side) heat-sink is considered as having a finite thermalcapacitance rate CH. The inlet and outlet temperatures of the cooling fluid are THin and THout , respectively. The low-temperature (cold-side) heat source is considered as having a finite thermal-capacitance rate CL. The inlet and outlet temperatures of the cooled fluid are TLin and TLout, respectively. The cycle is considered with an ideal gas having a constant thermal-capacitance rate (mass flow ratespecific heat product) Cwf. The hot- and cold- side heat-exchangers and the regenerator are considered, respectively, the counter-flow heat-exchangers, and their conductance {heat-transfer coefficient () and area (F) product} are UH ¼ H FH , UL ¼ L FL and UR ¼ R FR , respectively. Therefore, according to the properties of the heat-transfer processes, working fluid and heat-exchanger theory, the rate (QH) at which heat is rejected from the working fluid to heat sink, and the rate (QL) at which heat is transferred from the heat source to the working fluid, and the rate (QR) of the heat regenerated are, respectively, QH ¼ UH ½ðT3 THout Þ ðT6 THin Þ=ln½ðT3 THout Þ=ðT6 THin Þ ¼ Cwf ðT3 T6 Þ ¼ Cwf EH1 ðT3 THin Þ
ð2Þ
R ¼ QL ¼ UL ½ðTLin T5 Þ ðTLout T1 Þ=ln½ðTLin T5 Þ=ðTLout T1 Þ ¼ Cwf ðT5 T1 Þ ¼ Cwf EL1 ðTLin T1 Þ
ð3Þ
and QR ¼ Cwf ðT6 T4 Þ ¼ Cwf ðT2 T5 Þ ¼ Cwf ER ðT6 T5 Þ
ð4Þ
where EH1, EL1 and ER are the effectivenesses of the hot- and cold-side heatexchangers and the regenerator, respectively, and are defined as: EH1 ¼
1 exp½NH1 ð1 CHmin =CHmax Þ 1 ðCHmin =CHmax Þexp½NH1 ð1 CHmin =CHmax Þ
ð5Þ
EL1 ¼
1 exp½NL1 ð1 CLmin =CLmax Þ 1 ðCLmin =CLmax Þexp½NL1 ð1 CLmin =CLmax Þ
ð6Þ
ER ¼ NR =ð1 þ NR Þ
ð7Þ
where CHmin and CHmax are the minimum and maximum of CH and Cwf respectively, CLmin and CLmax are the minimum and maximum of CL and Cwf , and N is the number of heat-transfer units, respectively: NH1 ¼ UH =CHmin ; NL1 ¼ UL =CLmin ; NR ¼ UR =Cwf
ð8Þ
CHmin ¼ min CH ; Cwf ; CHmax ¼ max CH ; Cwf
ð9Þ
CLmin ¼ min CL ; Cwf ; CLmax ¼ max CL ; Cwf
ð10Þ
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3. Analytical formulae about cooling-load and cooling-load density Chen et al. [10] have derived the relationship between cooling load and COP as following: Cwf CLmin EL1 c Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 TLin ð1 ER Þ D m x1 t t þ 1 THin EH1 CHmin c R¼ 2 Cwf c ðx þ c 1Þ Cwf EH1 CHmin ER Cwf D m x1 t t þ 1 Cwf EL1 CLmin ER Cwf c þ ðx þ c 1Þ Cwf EH1 CHmin ð1 2ER Þ
1 þ "1
ð11Þ CHin EH1 ðx þ c 1Þð1 ER ÞCLmin EL1 TLin c Cwf ðx þ c 1Þ ER Cwf Cwf CLmin EL1 D m x1 t t þ 1 c ER þ ð1 2ER Þ ðx þ c 1Þ THin ¼ CLmin EL1 c Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 TLin ð1 ER Þ D m x1 t t þ 1 c CHmin EH1 THin ð12Þ
where x is the working-fluid temperature ratio of the cycle, that is, x ¼ T4 =T1s ¼ T3s =T2 ¼ m , where m ¼ ðk 1Þ=k, and k is the ratio of the specific heats. Fig. 1 shows that the specific volume at state 2 is the maximum in the cycle. The cooling-load density of the cycle is defined as r ¼ R=v2 . The dimensionless coolingload is written as R ¼ R=ðCL TLin Þ, and dimensionless cooling-load density is: r ¼ r=ðCL TLin =v1 Þ ¼ ðR=CL TLin Þ ðv1 =v2 Þ ¼ R ðDT1 =T2 Þ ¼ Cwf CLmin EL1 fc Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 ð1 ER Þ D m x1 t t þ 1 CHmin EH1 1 c g D m x1 t t þ 1 Cwf ER c þ ðx þ c 1Þ Cwf CHmin EH1 ð1 2ER Þ CLmin EL1 þ ð1 ER Þ CHmin EH1 Cwf 1 c 2 c CL ðx þ c 1Þ Cwf CHmin EH1 Cwf ER CL D m x1 t t þ 1 CL Cwf Cwf CLmin EL1 Cwf ER c þ ðx þ c 1Þ Cwf CHmin EH1 ð1 2ER Þ CHmin EH1 Cwf ER c 1 þ Cwf CLmin EL1 c ð1 ER Þ þ ð1 2ER Þ D m x1 t t þ 1 Cwf CLmin EL1 CHmin EH1 c 1 ð13Þ where 1 ¼ THin =TLin , is the inlet temperature ratio of the heat-reservoirs.
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4. Performance comparison under cooling-load density and cooling-load objective To see the advantages and disadvantages of the maximum cooling-load density design for the fixed heat-conductance of heat-exchangers, detailed numerical examples are provided and compared with those for the maximum cooling-load objective. In the calculations, k=1.4, 1=1.25, c =t =0.8, CL ¼ CH =1.0 W/K, Cwf =0.8 W/ K, EH1 ¼ EL1 =0.9 and D=0.96 are set. Fig. 2 shows the relationship of COP versus normalized dimensionless coolingload density (r/rmax; ) and normalized dimensionless cooling-load (R /R max; ) with ER=0.9, where rmax; is the maximum cooling-load density and R max; is the maximum cooling-load corresponding to the optimal pressure-ratio, respectively. Fig. 3 shows the comparison of the optimal COP ("r) corresponding to rmax; and the optimal COP ("R ) corresponding to R max; versus regenerator effectiveness (ER ). They show that "r is larger than "R . Fig. 4 shows the comparison of maximum specific volume (ðv2 =v1 Þr) of the cycle corresponding to maximum cooling-load density and that (ðv2 =v1 ÞR ) corresponding to maximum cooling-load. The figure shows that ðv2 =v1 Þr is smaller obviously than ðv2 =v1 ÞR . The calculations show that the coolingload density objective is better than the cooling load objective. The former objective optimization can achieve a higher COP and smaller air refrigerator size. Fig. 3 shows that "r="R when the regenerator effectiveness (ER ) equals zero, i.e., the cycle is a simple one, and "r > "R when ER > 0, i.e., the cycle becomes a regenerated one. The larger the ER is, the more obvious the difference. The analogous conclusion can be obtained for the maximum specific volume corresponding to the two optimization objectives above. The advantages of cooling-load density objective for the regenerated cycle are more obvious than those for the simple one.
Fig. 2. Normalized dimensionless cooling-load density and normalized dimensionless cooling-load versus COP.
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Fig. 3. The optimal COPs corresponding to maximum cooling-load density and maximum cooling-load versus regenerator effectiveness.
Fig. 4. The maximum cycle specific volume corresponding to maximum cooling-load density and maximum cooling-load versus regenerator effectiveness.
Fig. 5 shows a three-dimensional diagram for cooling load density (r), pressure ratio () and regenerator effectiveness (ER ). It shows that the curve of r versus is parabolic-like. They show that the increase of ER leads on an increase of r when the pressure ratio is smaller than a certain value. Chen et al. [10] have pointed out that R increases monotonously when both and ER increase, which is different from that of the cooling-load density objective. Fig. 6 shows the effect of heat-exchanger effectiveness (EH1 ¼ EL1 ) on the r versus with 1 =1.25, ER =0.9 and c =t =0.8. The figure shows that the heat-exchanger
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Fig. 5. Three-dimensional diagram of dimensionless cooling-load density, pressure ratio and regenerator effectiveness.
Fig. 6. The dimensionless cooling-load density versus pressure ratio and the effectiveness of the heatexchanger.
effectiveness does not affect the characteristic of the dimensionless cooling-load density versus pressure ratio (r) qualitatively. For different values of EH1 and EL1 , the curve of r versus maintains a parabolic-like shape. The effects of other parameters, such as 1 , D, c and t , on the relationships of r versus are similar to those of EH1 and EL1 .
5. Optimal distribution of heat conductance If the heat conductance of the heat exchangers are changeable, the dimensionless cooling-load density may be optimized by searching the optimal distribution for the
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fixed total heat-exchanger inventory. For the fixed heat-exchanger inventory UT , that is, for the constraint of UH þ UL þ UR ¼ UT , defining the hot-side heatconductance distribution (uH ) and the cold-side heat-conductance distribution (uL ) as the following: uH ¼ UH =UT ; uL ¼ UL =UT
ð14Þ
leads to: UH ¼ uH UT ; UL ¼ uL UT ; UR ¼ ð1 uH uL ÞUT
ð15Þ
The cooling-load density optimization is performed using numerical calculations. In these calculations, k=1.4 and Cwf =0.8 kW/K are set. The dimensionless coolingload density (r) and the COP (") of the cycle versus the hot-side heat-conductance distribution (uH ) and the cold-side heat-conductance distribution (uL ) with CL ¼ CH =1 kW/K, UT =5 kW/K, 1 =1.25, D=0.96, =10 and c =t =0.8 are shown in Figs. 7 and 8. The two three-dimensional diagrams show the existence of the optimal dimensionless cooling-load density and the optimal COP. One can see that for a fixed , there exists a pair of uHopt and uLopt , which lead to the optimal dimensionless cooling-load density (rmax;u ), and another pair of uHopt and uLopt , which lead to the optimal COP ("max ). In the vertical plane of the two diagrams, uH þ uL =1. If uH þ uL =1, the regenerated air refrigeration cycle becomes a simple one. If uH þ uL > 1, the practice plant does not exist. Therefore, in the numerical calculations, the following conditions should be satisfied: uH 4 1; uL 4 1; uH þ uL 4 1
ð16Þ
The influence of cycle heat-reservoir temperature-ratio () on the optimal dimension cooling-load density (rmax;u ) versus pressure ratio () with UT =5 kW/K, D=0.96, CL ¼ CH =1 kW/K and c =t =0.8 is shown in Fig. 9. The corresponding
Fig. 7. The dimensionless cooling-load density versus heat conductance distribution.
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Fig. 8. The COP versus heat-conductance distribution.
Fig. 9. The optimal cooling-load density versus pressure ratio and cycle heat-reservoir temperature ratio.
optimal heat-conductance distributions (uHopt and uLopt ) are shown in Figs. 10 and 11.The dashed section of the curves in Figs. 9–11 have no meaning in practice because they correspond to the condition of uH þ uL > 1. They are used only for one to observe the numerical potential. The calculations show there exist a pair of uHopt and uLopt , which lead to the optimal dimensionless cooling-load density (rmax;u ). The value of rmax;u increases when the cycle heat-reservoir temperature-ratio (1 ) decreases. When the pressure ratio () increases, rmax;u increases monotonously, and uHopt and uLopt increase, too.
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Fig. 10. The optimal hot-side heat-conductance distribution versus pressure ratio and cycle heat-reservoir temperature ratio.
Fig. 11. The cold-side heat-conductance distribution versus pressure ratio and cycle heat-reservoir temperature ratio.
An analogous study can be carried out for analyzing the effect of the heat inventory (UT ), the efficiencies of the compressor and expander (c =t ), and the pressurerecovery coefficients (D) on rmax;u . They show that the optimal dimensionless cooling-load density (rmax;u ) increases with increases in the total heat-exchanger inventory(UT ), the efficiencies of compressor and expander (c =t ), and the pressurerecovery coefficient (D). The optimal hot-side heat-conductance (uHopt ) increases with increases in the pressure-recovery coefficient (D) and the efficiencies of the compressor and expander (c =t ), decreases with increases in the total heat-exchanger inventory (UT ). The optimal cold-side heat-conductance (uLopt ) increases with increases in the
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total heat-exchanger inventory (UT ), the pressure-recovery coefficient (D) and the efficiencies of the compressor and expander (c =t ).
6. Optimal heat-capacity rate matching between the working fluid and heat reservoirs The effect of heat-capacity rate matching (Cwf =CH ) between the working fluid and the heat-reservoirs on the cooling-load density r is analyzed by numerical examples as mentioned previously. In the calculation, k ¼1.4, CL ¼1.2 kW/K, UT =5 kW/K,
Fig. 12. The cooling-load density versus heat-capacitance rate matching between the working fluid and heat-reservoirs and ratio of heat-reservoir capacitance rate.
Fig. 13. The cooling-load density versus heat-capacitance rate matching between the working fluid and heat-reservoirs and heat-exchanger inventory.
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D=0.96, =10, c =t =0.8 and 1 =1.25 are set. Fig. 12 shows the relationships of r versus Cwf =CH and ratio (CL =CH ) of the heat-reservoir capacitance rates. Fig. 13 shows the relationships of r versus Cwf =CH and heat inventory UT with CL =CH =1. The numerical examples show that the heat-capacity rate matching Cwf =CH between the working fluid and heat-reservoir influences the cooling-load density. The relationships are also affected by the ratio of the heat-reservoirs capacitance rate CL =CH . For the fixed CL =CH , the curve of r versus Cwf =CH is a parabolic-like shaped one. That is, there exists an optimal dimensionless cooling-load density rmax . When CL =CH increases, r decreases. The dimensionless cooling-load density r always increases when the total heat-exchanger inventory UT increases. But the increment is not obvious when UT becomes larger than a certain value. The effects of other parameters, such as 1 , D, and c and t , on the relationships of r versus Cwf =CH are similar to those of UT and CL =CH .
7. Conclusion The performance of a regenerated air refrigeration cycle with heat-transfer irreversibility in the hot- and cold-side heat-exchangers and the regenerator, irreversible compression and expansion losses in the compressor and expander, and a pressuredrop loss in the piping was examined. Comparisons between the maximum coolingload density performance and maximum cooling-load performance were carried out. The maximum cooling-load density design has the advantage of higher efficiencies and smaller size. Optimization was performed by searching the distribution among heat conductances of the hot- and cold-side exchangers and the regenerator for the fixed total heat-exchanger inventory, and by searching the optimal heat-capacitance rate matching between the working fluid and the heat-reservoirs on the cooling-load density. The influences of various parameters on the optimal cooling-load density and optimal distribution of heat inventory are analyzed. The analysis and optimization carried out in this paper include several special cases: (i) Regenerated cycle with constant-temperature reservoirs when CL ¼ CH ! 1. (ii) Irreversible simple cycle coupled to variable- or constant-temperature reservoirs when ER =0. (iii) Endoreversible simple cycle coupled to variable- or constant-temperature reservoirs when c ¼ t ¼ D1 ¼ D2 ¼ 1 and ER =0. (iv) Ideal simple (ER ¼ 0) or regenerated (ER 6¼ 0) reversible cycle when EH1 ¼ EL1 ¼ c ¼ t ¼ D1 ¼ D2 ¼ 1. The optimization will lead to a regenerated air refrigerator of smaller size and higher COP than those obtained by cooling-load optimization. The analysis and optimization may provide guidelines for the design of real air refrigerators.
Acknowledgements This paper is supported by the National Key Basic Research and Development Program of PR China (Project No. G2000026301), the Foundation for the Authors
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of National Excellent Doctoral Dissertations of PR China (Project No. 200136) and the Natural Science Foundation of the Naval University of Engineering, PR China.
References [1] Hamlin S, Hunt R, Tassou SA. Enhancing the performance of evaporative spray cooling in air cycle refrigeration and air condition technology. Appl Thermal Engng 1998;18(10):1139–48. [2] Mc Cormick JA. Progress on the development of miniature turbomachines for low capacity reverse Brayton cryocooler. Proc 9th Int Cryocooler Conf 1996;1996:255–67. [3] Saygin H, Sisman A. Brayton refrigeration cycles working under quantum degeneracy conditions. Appl Energy 2001;69(2):77–85. [4] Sieniutycz S, Salamon P. Advances in thermodynamics, vol. 4: finite-time thermodynamics and thermoeconomics. New York: Taylor & Francis; 1990. [5] Bejan A. Entropy generation minimization. Boca Raton FL: CRC Press; 1996. [6] Berry RS, Kazakov VA, Sieniutycz S, Szwast Z, Tsirlin AM. Thermodynamic optimization of finitetime processes. Chichester: Wiley; 1999. [7] Chen L, Wu C, Sun F. Finite-time thermodynamics optimization or entropy generation minimization of energy systems. J Non-Equilib Thermodyn 1999;24(3):327–59. [8] Chen L, Wu C, Sun F. Cooling-load versus COP characteristics for an irreversible air refrigeration cycle. Energy Convers Mgmt 1998;39(1/2):117–25. [9] Chen L, Wu C, Sun F. Finite-time thermodynamic performance of an isentropic closed regenerated Brayton refrigeration cycle. Int J Energy, Environment, Economics 1997;4(4):261–74. [10] Chen L, Zhou S, Sun F, Wu C, Performance of heat-transfer irreversible regenerated Brayton refrigerators J. Phys. D: Appl Phys 2001;34(5):830–7. [11] Luo J, Chen L, Sun F, Wu C. Optimum allocation of heat-exchanger inventory of irreversible air refrigeration cycles. Physica Scripta 2002;65(5):410–4. [12] Erbay LB, Yavuz H. The maximum cooling density of a realistic Stirling refrigerator. J Phys D: Appl Phys 1998;31(3):291–3. [13] Yavuz HL, Erbay LB, General performance characteristics of an Ericsson refrigerator. ECOS98, Nancy, France, 1998: 565-571. [14] Zhou S, Chen L, Sun F, Wu C. Cooling-load density analysis and optimization for an endoreversible air refrigerator. Open Sys Information Dyn 2001;8(2):147–55. [15] Zhou S, Chen L, Sun F, Wu C. Cooling-load density characteristics of an endoreversible variabletemperature heat-reservoir air refrigerator. Int J Energy Res 2002;26(10):881–92. [16] Zhou S, Chen L, Sun F, Wu C. Cooling-load density optimization of an irreversible simple Brayton refrigerator. Open Sys Information Dyn 2002;9(4):325–37.
Cooling-load density optimization for a regenerated air refrigerator Shengbing Zhoua, Lingen Chena,*, Fengrui Suna, Chih Wub a Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Mechanical Engineering Department, US Naval Academy, Annapolis, MD 21402, USA
b
Accepted 21 August 2003
Abstract A performance analysis and optimization of a regenerated air refrigeration cycle with variable-temperature heat-reservoirs is carried out by taking the cooling-load density, i.e., the ratio of cooling load to the maximum specific volume in the cycle, as the optimization objective using finite-time thermodynamics (FTT) or entropy-generation minimization (EGM). The model of a regenerated air refrigerator is presented, and analytical relationships between cooling-load density and pressure ratio, as well as between coefficient of performance (COP) and pressure ratio are derived. The irreversibilities considered in the analysis include the heattransfer losses in the hot- and cold-side heat-exchangers and the regenerator, the non-isentropic compression and expansion losses in the compressor and expander, and the pressuredrop losses in the piping. The cycle performance comparison under maximum cooling-load density and maximum cooling-load conditions is performed via detailed numerical calculations. The optimal performance characteristics of the cycle are obtained by optimizing the pressure ratio of the compressor, and searching for the optimum distribution of heat-conductances of the hot- and cold-side heat-exchangers and regenerator for the fixed total heatexchanger inventory. The effect of heat capacity rate matching between the working fluid and heat reservoirs on the cooling-load density is analyzed for the cycle. The influences of the effectiveness of the regenerator as well as the hot- and cold-side heat-exchangers, the efficiencies of the expander and the compressor, the pressure-recovery coefficient, and the temperature ratio of the heat reservoirs on the cooling-load density and COP are examined and illustrated by numerical examples. # 2003 Elsevier Ltd. All rights reserved. Keywords: Regenerated air refrigeration cycle; Variable-temperature heat reservoirs; Cooling-load density; Optimization
* Corresponding author. Tel.: 0086-27-83615046; fax: 0086-27-83638709. E-mail address: [email protected], [email protected] (L. Chen). 0306-2619/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2003.08.008
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1. Introduction Research upon air refrigeration cycles, in which air is the working fluid, has been paid more attention recently due to the destruction of the Ozonosphere by chlorofluorocarbon (CFC) and the pressure of environmental protection [1–3]. Optimizations of air cycles are also carried out using finite-time thermodynamics (FTT) or entropy-generation minimization (EGM) [4–7]. Chen et al. investigated the cooling-load versus COP characteristics of a simple [8] and a regenerated [9,10] air refrigeration cycle with heat-transfer loss and/or other irreversibilities. Luo et al. [11] optimized the cooling-load and the COP of a simple irreversible air refrigeration cycle by searching for the optimum pressure-ratio of the compressor and the optimum distribution of heat conductance of the hot- and cold-side heat-exchangers for the fixed total heat-exchanger inventory. Some authors, including Erbay and Yavuz [12], and Yavuz and Erbay [13], have taken the cooling-load density, defined as the ratio of cooling-load to the maximum specific volume in the cycle, as the optimization objective and analyzed the optimum performance of the Stirling refrigerator as well as the Ericsson refrigerator with a realistic regenerator due to the polytropic processes applied for the heat absorption and rejection. Zhou et al. analyzed and optimized the cooling-load density of endoreversible simple air refrigeration cycles coupled to constant- [14] and variable[15] temperature heat-reservoirs, and of an irreversible simple air refrigeration cycle coupled to constant-temperature heat-reservoirs [16]. The aim of this paper is to analyze and optimize the performance of a regenerated air refrigeration cycle with variable-temperature heat-reservoirs, which is much closer to the real air refrigerator, using a cooling-load density objective based on the model presented by Chen et al. [14]. The analytical formula about cooling-load density is derived by taking into account heat-resistance losses in the hot- and coldside heat-exchangers, the non-isentropic expansion and compression losses in the expander and compressor, and the pressure-drop loss in the piping. The coolingload density objective is to optimize the cycle performance including the refrigerator size effects.
2. Air refrigerator cycle A regenerated air refrigeration cycle with variable-temperature heat-reservoirs and its surroundings, to be considered in this paper, are shown in Fig. 1(a) and (b). Between 2 and 3 in the TS diagram is a non-isentropic compression process in the compressor. Process 3-6 is heat rejection to the heat sink in the hot-side heat-exchanger. Process 6-4 is heat rejection in the regenerator. Process 4-1 is non- isentropic expansion process in the expander. Process 1-5 is a heat addition (cooling) process from the heat source in the cold-side heat-exchanger. Process 5-2 is the heat addition process in the regenerator. Processes 2-3s and 4-1s are isentropic compression and expansion in the ideal compressor and expander corresponding to the non-isentropic processes 2-3 and 4-1, respectively. The compressor and expansion efficiencies are defined as:
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Fig. 1. (a) The schematic of the regenerated-air refrigerator A—the latter cooler; B—the refrigerator; C—the compressor; R—the regenerator; T—the turbine cooler (expander). (b) The temperature–entropy diagram of the regenerated-air refrigeration cycle.
c ¼ ðT3s T2 Þ=ðT3 T2 Þ; t ¼ ðT4 T1 Þ=ðT4 T1s Þ
ð1Þ
The pressure ratio of the compressor is ¼ P3 =P2 , where P is the pressure. The pressure-drop P in the piping is expressed by the pressure recovery coefficient D ¼ ð1 DP12 =P1 Þð1 DP34 =P4 Þ. The pressure ratio of the expansion process is P4 =P1 ¼ DP3 =P2 ¼ D.
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The high-temperature (hot-side) heat-sink is considered as having a finite thermalcapacitance rate CH. The inlet and outlet temperatures of the cooling fluid are THin and THout , respectively. The low-temperature (cold-side) heat source is considered as having a finite thermal-capacitance rate CL. The inlet and outlet temperatures of the cooled fluid are TLin and TLout, respectively. The cycle is considered with an ideal gas having a constant thermal-capacitance rate (mass flow ratespecific heat product) Cwf. The hot- and cold- side heat-exchangers and the regenerator are considered, respectively, the counter-flow heat-exchangers, and their conductance {heat-transfer coefficient () and area (F) product} are UH ¼ H FH , UL ¼ L FL and UR ¼ R FR , respectively. Therefore, according to the properties of the heat-transfer processes, working fluid and heat-exchanger theory, the rate (QH) at which heat is rejected from the working fluid to heat sink, and the rate (QL) at which heat is transferred from the heat source to the working fluid, and the rate (QR) of the heat regenerated are, respectively, QH ¼ UH ½ðT3 THout Þ ðT6 THin Þ=ln½ðT3 THout Þ=ðT6 THin Þ ¼ Cwf ðT3 T6 Þ ¼ Cwf EH1 ðT3 THin Þ
ð2Þ
R ¼ QL ¼ UL ½ðTLin T5 Þ ðTLout T1 Þ=ln½ðTLin T5 Þ=ðTLout T1 Þ ¼ Cwf ðT5 T1 Þ ¼ Cwf EL1 ðTLin T1 Þ
ð3Þ
and QR ¼ Cwf ðT6 T4 Þ ¼ Cwf ðT2 T5 Þ ¼ Cwf ER ðT6 T5 Þ
ð4Þ
where EH1, EL1 and ER are the effectivenesses of the hot- and cold-side heatexchangers and the regenerator, respectively, and are defined as: EH1 ¼
1 exp½NH1 ð1 CHmin =CHmax Þ 1 ðCHmin =CHmax Þexp½NH1 ð1 CHmin =CHmax Þ
ð5Þ
EL1 ¼
1 exp½NL1 ð1 CLmin =CLmax Þ 1 ðCLmin =CLmax Þexp½NL1 ð1 CLmin =CLmax Þ
ð6Þ
ER ¼ NR =ð1 þ NR Þ
ð7Þ
where CHmin and CHmax are the minimum and maximum of CH and Cwf respectively, CLmin and CLmax are the minimum and maximum of CL and Cwf , and N is the number of heat-transfer units, respectively: NH1 ¼ UH =CHmin ; NL1 ¼ UL =CLmin ; NR ¼ UR =Cwf
ð8Þ
CHmin ¼ min CH ; Cwf ; CHmax ¼ max CH ; Cwf
ð9Þ
CLmin ¼ min CL ; Cwf ; CLmax ¼ max CL ; Cwf
ð10Þ
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3. Analytical formulae about cooling-load and cooling-load density Chen et al. [10] have derived the relationship between cooling load and COP as following: Cwf CLmin EL1 c Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 TLin ð1 ER Þ D m x1 t t þ 1 THin EH1 CHmin c R¼ 2 Cwf c ðx þ c 1Þ Cwf EH1 CHmin ER Cwf D m x1 t t þ 1 Cwf EL1 CLmin ER Cwf c þ ðx þ c 1Þ Cwf EH1 CHmin ð1 2ER Þ
1 þ "1
ð11Þ CHin EH1 ðx þ c 1Þð1 ER ÞCLmin EL1 TLin c Cwf ðx þ c 1Þ ER Cwf Cwf CLmin EL1 D m x1 t t þ 1 c ER þ ð1 2ER Þ ðx þ c 1Þ THin ¼ CLmin EL1 c Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 TLin ð1 ER Þ D m x1 t t þ 1 c CHmin EH1 THin ð12Þ
where x is the working-fluid temperature ratio of the cycle, that is, x ¼ T4 =T1s ¼ T3s =T2 ¼ m , where m ¼ ðk 1Þ=k, and k is the ratio of the specific heats. Fig. 1 shows that the specific volume at state 2 is the maximum in the cycle. The cooling-load density of the cycle is defined as r ¼ R=v2 . The dimensionless coolingload is written as R ¼ R=ðCL TLin Þ, and dimensionless cooling-load density is: r ¼ r=ðCL TLin =v1 Þ ¼ ðR=CL TLin Þ ðv1 =v2 Þ ¼ R ðDT1 =T2 Þ ¼ Cwf CLmin EL1 fc Cwf 1 ER D m x1 t t þ 1 ðx þ c 1Þ Cwf CHmin EH1 ER þ ð1 2ER Þ D m x1 t t þ 1 ð1 ER Þ D m x1 t t þ 1 CHmin EH1 1 c g D m x1 t t þ 1 Cwf ER c þ ðx þ c 1Þ Cwf CHmin EH1 ð1 2ER Þ CLmin EL1 þ ð1 ER Þ CHmin EH1 Cwf 1 c 2 c CL ðx þ c 1Þ Cwf CHmin EH1 Cwf ER CL D m x1 t t þ 1 CL Cwf Cwf CLmin EL1 Cwf ER c þ ðx þ c 1Þ Cwf CHmin EH1 ð1 2ER Þ CHmin EH1 Cwf ER c 1 þ Cwf CLmin EL1 c ð1 ER Þ þ ð1 2ER Þ D m x1 t t þ 1 Cwf CLmin EL1 CHmin EH1 c 1 ð13Þ where 1 ¼ THin =TLin , is the inlet temperature ratio of the heat-reservoirs.
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4. Performance comparison under cooling-load density and cooling-load objective To see the advantages and disadvantages of the maximum cooling-load density design for the fixed heat-conductance of heat-exchangers, detailed numerical examples are provided and compared with those for the maximum cooling-load objective. In the calculations, k=1.4, 1=1.25, c =t =0.8, CL ¼ CH =1.0 W/K, Cwf =0.8 W/ K, EH1 ¼ EL1 =0.9 and D=0.96 are set. Fig. 2 shows the relationship of COP versus normalized dimensionless coolingload density (r/rmax; ) and normalized dimensionless cooling-load (R /R max; ) with ER=0.9, where rmax; is the maximum cooling-load density and R max; is the maximum cooling-load corresponding to the optimal pressure-ratio, respectively. Fig. 3 shows the comparison of the optimal COP ("r) corresponding to rmax; and the optimal COP ("R ) corresponding to R max; versus regenerator effectiveness (ER ). They show that "r is larger than "R . Fig. 4 shows the comparison of maximum specific volume (ðv2 =v1 Þr) of the cycle corresponding to maximum cooling-load density and that (ðv2 =v1 ÞR ) corresponding to maximum cooling-load. The figure shows that ðv2 =v1 Þr is smaller obviously than ðv2 =v1 ÞR . The calculations show that the coolingload density objective is better than the cooling load objective. The former objective optimization can achieve a higher COP and smaller air refrigerator size. Fig. 3 shows that "r="R when the regenerator effectiveness (ER ) equals zero, i.e., the cycle is a simple one, and "r > "R when ER > 0, i.e., the cycle becomes a regenerated one. The larger the ER is, the more obvious the difference. The analogous conclusion can be obtained for the maximum specific volume corresponding to the two optimization objectives above. The advantages of cooling-load density objective for the regenerated cycle are more obvious than those for the simple one.
Fig. 2. Normalized dimensionless cooling-load density and normalized dimensionless cooling-load versus COP.
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Fig. 3. The optimal COPs corresponding to maximum cooling-load density and maximum cooling-load versus regenerator effectiveness.
Fig. 4. The maximum cycle specific volume corresponding to maximum cooling-load density and maximum cooling-load versus regenerator effectiveness.
Fig. 5 shows a three-dimensional diagram for cooling load density (r), pressure ratio () and regenerator effectiveness (ER ). It shows that the curve of r versus is parabolic-like. They show that the increase of ER leads on an increase of r when the pressure ratio is smaller than a certain value. Chen et al. [10] have pointed out that R increases monotonously when both and ER increase, which is different from that of the cooling-load density objective. Fig. 6 shows the effect of heat-exchanger effectiveness (EH1 ¼ EL1 ) on the r versus with 1 =1.25, ER =0.9 and c =t =0.8. The figure shows that the heat-exchanger
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Fig. 5. Three-dimensional diagram of dimensionless cooling-load density, pressure ratio and regenerator effectiveness.
Fig. 6. The dimensionless cooling-load density versus pressure ratio and the effectiveness of the heatexchanger.
effectiveness does not affect the characteristic of the dimensionless cooling-load density versus pressure ratio (r) qualitatively. For different values of EH1 and EL1 , the curve of r versus maintains a parabolic-like shape. The effects of other parameters, such as 1 , D, c and t , on the relationships of r versus are similar to those of EH1 and EL1 .
5. Optimal distribution of heat conductance If the heat conductance of the heat exchangers are changeable, the dimensionless cooling-load density may be optimized by searching the optimal distribution for the
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fixed total heat-exchanger inventory. For the fixed heat-exchanger inventory UT , that is, for the constraint of UH þ UL þ UR ¼ UT , defining the hot-side heatconductance distribution (uH ) and the cold-side heat-conductance distribution (uL ) as the following: uH ¼ UH =UT ; uL ¼ UL =UT
ð14Þ
leads to: UH ¼ uH UT ; UL ¼ uL UT ; UR ¼ ð1 uH uL ÞUT
ð15Þ
The cooling-load density optimization is performed using numerical calculations. In these calculations, k=1.4 and Cwf =0.8 kW/K are set. The dimensionless coolingload density (r) and the COP (") of the cycle versus the hot-side heat-conductance distribution (uH ) and the cold-side heat-conductance distribution (uL ) with CL ¼ CH =1 kW/K, UT =5 kW/K, 1 =1.25, D=0.96, =10 and c =t =0.8 are shown in Figs. 7 and 8. The two three-dimensional diagrams show the existence of the optimal dimensionless cooling-load density and the optimal COP. One can see that for a fixed , there exists a pair of uHopt and uLopt , which lead to the optimal dimensionless cooling-load density (rmax;u ), and another pair of uHopt and uLopt , which lead to the optimal COP ("max ). In the vertical plane of the two diagrams, uH þ uL =1. If uH þ uL =1, the regenerated air refrigeration cycle becomes a simple one. If uH þ uL > 1, the practice plant does not exist. Therefore, in the numerical calculations, the following conditions should be satisfied: uH 4 1; uL 4 1; uH þ uL 4 1
ð16Þ
The influence of cycle heat-reservoir temperature-ratio () on the optimal dimension cooling-load density (rmax;u ) versus pressure ratio () with UT =5 kW/K, D=0.96, CL ¼ CH =1 kW/K and c =t =0.8 is shown in Fig. 9. The corresponding
Fig. 7. The dimensionless cooling-load density versus heat conductance distribution.
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Fig. 8. The COP versus heat-conductance distribution.
Fig. 9. The optimal cooling-load density versus pressure ratio and cycle heat-reservoir temperature ratio.
optimal heat-conductance distributions (uHopt and uLopt ) are shown in Figs. 10 and 11.The dashed section of the curves in Figs. 9–11 have no meaning in practice because they correspond to the condition of uH þ uL > 1. They are used only for one to observe the numerical potential. The calculations show there exist a pair of uHopt and uLopt , which lead to the optimal dimensionless cooling-load density (rmax;u ). The value of rmax;u increases when the cycle heat-reservoir temperature-ratio (1 ) decreases. When the pressure ratio () increases, rmax;u increases monotonously, and uHopt and uLopt increase, too.
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Fig. 10. The optimal hot-side heat-conductance distribution versus pressure ratio and cycle heat-reservoir temperature ratio.
Fig. 11. The cold-side heat-conductance distribution versus pressure ratio and cycle heat-reservoir temperature ratio.
An analogous study can be carried out for analyzing the effect of the heat inventory (UT ), the efficiencies of the compressor and expander (c =t ), and the pressurerecovery coefficients (D) on rmax;u . They show that the optimal dimensionless cooling-load density (rmax;u ) increases with increases in the total heat-exchanger inventory(UT ), the efficiencies of compressor and expander (c =t ), and the pressurerecovery coefficient (D). The optimal hot-side heat-conductance (uHopt ) increases with increases in the pressure-recovery coefficient (D) and the efficiencies of the compressor and expander (c =t ), decreases with increases in the total heat-exchanger inventory (UT ). The optimal cold-side heat-conductance (uLopt ) increases with increases in the
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total heat-exchanger inventory (UT ), the pressure-recovery coefficient (D) and the efficiencies of the compressor and expander (c =t ).
6. Optimal heat-capacity rate matching between the working fluid and heat reservoirs The effect of heat-capacity rate matching (Cwf =CH ) between the working fluid and the heat-reservoirs on the cooling-load density r is analyzed by numerical examples as mentioned previously. In the calculation, k ¼1.4, CL ¼1.2 kW/K, UT =5 kW/K,
Fig. 12. The cooling-load density versus heat-capacitance rate matching between the working fluid and heat-reservoirs and ratio of heat-reservoir capacitance rate.
Fig. 13. The cooling-load density versus heat-capacitance rate matching between the working fluid and heat-reservoirs and heat-exchanger inventory.
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D=0.96, =10, c =t =0.8 and 1 =1.25 are set. Fig. 12 shows the relationships of r versus Cwf =CH and ratio (CL =CH ) of the heat-reservoir capacitance rates. Fig. 13 shows the relationships of r versus Cwf =CH and heat inventory UT with CL =CH =1. The numerical examples show that the heat-capacity rate matching Cwf =CH between the working fluid and heat-reservoir influences the cooling-load density. The relationships are also affected by the ratio of the heat-reservoirs capacitance rate CL =CH . For the fixed CL =CH , the curve of r versus Cwf =CH is a parabolic-like shaped one. That is, there exists an optimal dimensionless cooling-load density rmax . When CL =CH increases, r decreases. The dimensionless cooling-load density r always increases when the total heat-exchanger inventory UT increases. But the increment is not obvious when UT becomes larger than a certain value. The effects of other parameters, such as 1 , D, and c and t , on the relationships of r versus Cwf =CH are similar to those of UT and CL =CH .
7. Conclusion The performance of a regenerated air refrigeration cycle with heat-transfer irreversibility in the hot- and cold-side heat-exchangers and the regenerator, irreversible compression and expansion losses in the compressor and expander, and a pressuredrop loss in the piping was examined. Comparisons between the maximum coolingload density performance and maximum cooling-load performance were carried out. The maximum cooling-load density design has the advantage of higher efficiencies and smaller size. Optimization was performed by searching the distribution among heat conductances of the hot- and cold-side exchangers and the regenerator for the fixed total heat-exchanger inventory, and by searching the optimal heat-capacitance rate matching between the working fluid and the heat-reservoirs on the cooling-load density. The influences of various parameters on the optimal cooling-load density and optimal distribution of heat inventory are analyzed. The analysis and optimization carried out in this paper include several special cases: (i) Regenerated cycle with constant-temperature reservoirs when CL ¼ CH ! 1. (ii) Irreversible simple cycle coupled to variable- or constant-temperature reservoirs when ER =0. (iii) Endoreversible simple cycle coupled to variable- or constant-temperature reservoirs when c ¼ t ¼ D1 ¼ D2 ¼ 1 and ER =0. (iv) Ideal simple (ER ¼ 0) or regenerated (ER 6¼ 0) reversible cycle when EH1 ¼ EL1 ¼ c ¼ t ¼ D1 ¼ D2 ¼ 1. The optimization will lead to a regenerated air refrigerator of smaller size and higher COP than those obtained by cooling-load optimization. The analysis and optimization may provide guidelines for the design of real air refrigerators.
Acknowledgements This paper is supported by the National Key Basic Research and Development Program of PR China (Project No. G2000026301), the Foundation for the Authors
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of National Excellent Doctoral Dissertations of PR China (Project No. 200136) and the Natural Science Foundation of the Naval University of Engineering, PR China.
References [1] Hamlin S, Hunt R, Tassou SA. Enhancing the performance of evaporative spray cooling in air cycle refrigeration and air condition technology. Appl Thermal Engng 1998;18(10):1139–48. [2] Mc Cormick JA. Progress on the development of miniature turbomachines for low capacity reverse Brayton cryocooler. Proc 9th Int Cryocooler Conf 1996;1996:255–67. [3] Saygin H, Sisman A. Brayton refrigeration cycles working under quantum degeneracy conditions. Appl Energy 2001;69(2):77–85. [4] Sieniutycz S, Salamon P. Advances in thermodynamics, vol. 4: finite-time thermodynamics and thermoeconomics. New York: Taylor & Francis; 1990. [5] Bejan A. Entropy generation minimization. Boca Raton FL: CRC Press; 1996. [6] Berry RS, Kazakov VA, Sieniutycz S, Szwast Z, Tsirlin AM. Thermodynamic optimization of finitetime processes. Chichester: Wiley; 1999. [7] Chen L, Wu C, Sun F. Finite-time thermodynamics optimization or entropy generation minimization of energy systems. J Non-Equilib Thermodyn 1999;24(3):327–59. [8] Chen L, Wu C, Sun F. Cooling-load versus COP characteristics for an irreversible air refrigeration cycle. Energy Convers Mgmt 1998;39(1/2):117–25. [9] Chen L, Wu C, Sun F. Finite-time thermodynamic performance of an isentropic closed regenerated Brayton refrigeration cycle. Int J Energy, Environment, Economics 1997;4(4):261–74. [10] Chen L, Zhou S, Sun F, Wu C, Performance of heat-transfer irreversible regenerated Brayton refrigerators J. Phys. D: Appl Phys 2001;34(5):830–7. [11] Luo J, Chen L, Sun F, Wu C. Optimum allocation of heat-exchanger inventory of irreversible air refrigeration cycles. Physica Scripta 2002;65(5):410–4. [12] Erbay LB, Yavuz H. The maximum cooling density of a realistic Stirling refrigerator. J Phys D: Appl Phys 1998;31(3):291–3. [13] Yavuz HL, Erbay LB, General performance characteristics of an Ericsson refrigerator. ECOS98, Nancy, France, 1998: 565-571. [14] Zhou S, Chen L, Sun F, Wu C. Cooling-load density analysis and optimization for an endoreversible air refrigerator. Open Sys Information Dyn 2001;8(2):147–55. [15] Zhou S, Chen L, Sun F, Wu C. Cooling-load density characteristics of an endoreversible variabletemperature heat-reservoir air refrigerator. Int J Energy Res 2002;26(10):881–92. [16] Zhou S, Chen L, Sun F, Wu C. Cooling-load density optimization of an irreversible simple Brayton refrigerator. Open Sys Information Dyn 2002;9(4):325–37.