Optimal cooling load and COP relationship of a four-heat-reservoir endoreversible absorption refrigeration cycle. Entropy 2004

Abstract: On the basis of a four-heat-reservoir endoreversible absorption refrigeration cycle model, another linear heat transfer law [i.e., the heat-flux Q ∝ ∆( T −1 ) ] is adopted, the fundamental optimal relation between the coefficient of performance (COP) and the cooling load, as well as the maximum cooling load and the corresponding COP of the cycle coupled to constant-temperature heat reservoirs are derived by using finite-time thermodynamics or thermodynamic optimization. The optimal distribution of the heat-transfer surface areas is also obtained. Moreover, the effects of the cycle parameters on the COP and the cooling load of the cycle are studied by detailed numerical examples. The results obtained herein are of importance to the optimal design and performance improvement of an absorption refrigeration cycle. Keywords: Finite Time Thermodynamics, Thermodynamic Optimization, Four-Heat- Reservoir Endoreversible Absorption Refrigeration Cycle, Optimal Performance __________________________________________________________________________________


Introduction
The absorption refrigerators can be driven by 'low-grade' heat energy such as waste heat in industries, solar energy and geothermal energy, and have a large potential for reducing the heat pollution for the environment.Thus, absorption refrigerators for industrial and domestic use are generating renewed interest throughout the world.In the last years, finite-time thermodynamics (or endoreversible thermodynamics, or entropy generation minimization, or thermodynamic optimization) [1][2][3][4] was applied to the performance study of absorption refrigerators, and a lot of results, which are different from those by using the classical thermodynamics, have been obtained.Yan et al. [6], Wijeysundera [7,8], Goktun [9] and Chen et al. [10] analyzed the performance of the three-heatreservoir endoreversible [6][7][8] and irreversible [9,10] absorption refrigeration cycles with Newton's heat transfer law.Chen et al. [11][12][13] studied the performance of the three-heat-reservoir endoreversible [11,12] and irreversible [13] absorption refrigeration cycles with another linear heat transfer law, i.e., linear phenomenological law, . A three-heat-reservoir absorption refrigerator is a simplified model that the temperature of a condenser is equal to that of an absorber, but a real absorption refrigerator is not.Therefore, a four-heat-reservoir absorption refrigeration cycle model is closer to a real absorption refrigerator.The performance of the four-heat-reservoir absorption refrigeration cycle with Newton's heat transfer law was studied by Chen [14], Shi et al. [15] and Zheng et al. [16].Chen [14] deduced the maximum cooling load limit and the corresponding coefficient of performance (COP) of the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer, Shi et al. [15] deduced the fundamental optimal relation between the cooling load and the COP of the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer, and Zheng et al. [16] deduced the optimal heat transfer surface areas of the four heat exchangers the endoreversible four-heat-reservoir absorption refrigeration cycle with the sole irreversibility of heat transfer.On the basis of these research work, a four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law is established in this paper.The fundamental optimal relation between the COP and the cooling load, as well as the maximum cooling load and the corresponding COP of the cycle are derived.The results can provide the theoretical bases for the optimal design and operation of real absorption refrigerator operating between four temperature levels.The present work is different from a recent work of the authors [17].In Ref. [17], an endoreversible four-heat-reservoir absorption heat-transformer with Newton's heat transfer law was established, and the fundamental optimal relation between the COP and the heating load, as well as the maximum heating load and the corresponding COP of the cycle were derived.

Physical Model
A four-heat-reservoir endoreversible absorption refrigeration cycle that consists of a generator, an evaporator, an absorber and a condenser is shown in Fig. 1.The flow of the working fluid in the cycle system is stable and the different parts of the working fluid exchange heat with the heat reservoirs at temperature H T , L T , O T and M T during the full time (cycle period) τ , whereas there are thermal resistances between the working fluid and the external heat reservoirs.Therefore, the corresponding working fluid temperatures are 1 T , 2 T , 3 T and 4 T , respectively.Work input required by the solution pump in the system is negligible relative to the energy input to the generator and is often neglected for the purpose of analysis [6][7][8][9][10][11][12][13][14][15][16][17].It is assumed that the heat transfers between the working fluid in the heat exchangers and the external heat reservoirs are carried out under a finite temperature difference and obey linear phenomenological heat transfer law [i.e., the heat-flux ], and these heat exchange processes are isothermal and the equations of heat transfer may be written as U and 4 U are, respectively, the overall heat-transfer coefficients of the generator, evaporator, condenser and absorber; and 1 A , 2 A , 3 A and 4 A are, respectively, the heat-transfer surface areas of the generator, evaporator, condenser and absorber.The overall heat-transfer surface area A is

Fundamental Optimal Relation
From the first law of thermodynamics, one has From the second law of thermodynamics and the endoreversible property of the cycle, one has Defining the parameter a (the distribution of the total heat reject quantity between the condenser and the absorber) From equations ( 6) and (7), one can obtain the COP of an endoreversible absorption refrigerator Using equations ( 1) -( 9), the cooling load of the refrigerator can be written as Using equation (10) and the extremal conditions , one can derive the temperatures of the working fluid in the generator, absorber, condenser and evaporator, which correspond the optimal cooling load for the given COP.Substituting them into equation ( 10) yields the fundamental optimal relation between the cooling load and the COP of the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law as follows Equation ( 11) is the major results of this paper.It can reveal the ε − R characteristics of a four-heat-reservoir endoreversible absorption refrigeration cycle affected by thermal resistance, and some significant results and new bounds may be derived from it.
Using equations ( 1)-( 4), (11), and the temperatures of the working fluid in the generator, absorber, condenser and evaporator which are derived, one can obtain the optimal distribution relation of the heat-transfer surface areas as follows From equations ( 5) and ( 12)-( 14), one can find the relations between the heat-transfer surface areas of each heated exchanger and the total heat-transfer surface area A as follows   It can be seen that the optimal COP of the four-heat-reservoir endoreversible absorption refrigeration cycle can't exceed the reversible COP r ε .This shows that the real absorption refrigerators must decrease the COP level if one wants to obtain some cooling load.2. When r ε ε < , there exists a maximum cooling load.Using equation ( 11) and the extremal condition where m R is the maximum cooling load, and R ε is the corresponding COP.
, the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law becomes the three-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law, and equations ( 20) and (21) become [11] of the temperatures of the condenser to the evaporator, and the ratio of the temperatures of the generator to the absorption, one can analyze the performance numerically.In the calculation, are set [5, 18] .The influence of a on the optimal cooling load R versus the COP ε of the four-heat-reservoir endoreversible absorption refrigeration cycle with is shown in Fig. 2. The influence of x on the optimal cooling load R versus the COP ε with ε and R ε increase with the increase of a .These Fig. 2 The influence of a on the optimal cooling load R versus the COP ε Fig. 3 The influence of x on the optimal cooling load R versus the COP ε Fig. 4 The influence of y on the optimal cooling load R versus the COP ε

Conclusion
The performance of the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law is analyzed and optimized by using finite-time thermodynamics in this paper.Moreover, the effects of the cycle parameters on the COP and the cooling load of the cycle are studied by detailed numerical examples.The selection range for the practice parameters of the four-heat-reservoir endoreversible absorption refrigeration cycle with another linear heat transfer law are derived.The results obtain herein have realistic significance and may provide some new theoretical guidance for the optimal design and performance improvement of real absorption refrigerators.

Fig. 1 A
Fig.1 A four-heat-reservoir endoreversible absorption cycle model obtain the reversible COP r ε of the four-heat- m R and R ε are two important parameters of the four-heat-reservoir endoreversible absorption refrigeration cycle with linear phenomenological heat transfer law, because they determine the upper bound for the cooling load and the lower bound of the COP, and provide a finite-time thermodynamic criteria for the optimal design of real absorption refrigerators, i.e., the real absorption refrigerator design must match the condition

. 3 .. 4 .
The influence of y on the optimal cooling load R versus the COP ε with Figures2-4show that the optimal cooling load R versus the COP ε is a parabolic curve, and there exists a maximum cooling load m R and the corresponding COP R ε .When m R R < , there exist two different ε for a fixed R , one is larger than R ε , and another is smaller than R ε .When R ε ε < , the COP decreases with the decrease of the cooling load, so the optimal operation range of the COP of the absorption refrigerators should be selected in x on the reversible COP r ε , the maximum cooling load m R and the corresponding COP R ε versus a with Figs.5-7.Figures5-7show that for a fixed a , r ε , m R and R ε decrease with the increase of x .When a is larger than one in value, the influence of the r ε , m R and R ε are less.There is a special point 1172