Efect of heat transfer law on the finite-time exergoeconomic performance of a Carnot refrigerator

a b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F K F1 Z2 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ß a 295 Exergy Int. J. 1(4) (2001) 295–302 www.exergyonline.com E?ect of heat transfer law on the ?nite-time exergoeconomic performance of a Carnot refrigerator Lingen Chen a , Chih Wu b* , Fengrui Sun a Faculty 306, Naval University of Engineering, Wuhan 430033, People’s Republic of China Department of Mechanical Engineering, U.S. Naval Academy, Annapolis, MD 21402, USA (Received 6 June 1999, accepted 15 May 2000) Abstract — The operation of a Carnot refrigerator is viewed as a production process with exergy as its output. The economic optimization of the endoreversible refrigerator is carried out in this paper. The Coe?cient of Performance (COP) of the refrigerator is a secondary consideration of the practical engineering e?ort of maximizing cooling rate and exergy whose goodness is constrained by economical considerations. Therefore, the pro?t of the refrigerator is taken as the optimization objective. Using the method of ?nite-time exergoeconomic analysis, which emphasizes the compromise optimization between economics (pro?t) and the appropriate energy utilization factor (Coe?cient of Performance, COP) for ?nite-time (endoreversible) thermodynamic cycles, this paper derives the relation between optimal pro?t and COP of an endoreversible Carnot refrigerator based on a relatively general heat transfer law q ? (T n ). The COP at the maximum pro?t is also obtained. The results obtained involve those for three common heat transfer laws: Newton’s law (n = 1), the linear phenomenological law in irreversible thermodynamics (n = -1), and the radiative heat transfer law (n = 4). ? 2001 Éditions scienti?ques et médicales Elsevier SAS Nomenclature T T0 temperature . . . . . . . . . . . environmental temperature . . . . . . . . . . . . . . . . . . . . . . . . K . K A C D E E1 exergy output of refrigerator . . . . . . . . . cost . . . . . . . . . . . . . . . . . . . . . . . function de?ned in equation (31) . . . . . . . function de?ned in equation (29) . . . . . . . function de?ned in equation (68) function de?ned in equation (29) . . . . . . . . . . . kJ . . $ ·s-1 K·kW-1 . . . . K . . . . TH TL TWH TWL W Z1 heat sink temperature . . . . . . heat source temperature . . . . . warm refrigerant temperature . . cold refrigerant temperature . . work . . . . . . . . . . . . . . . function de?ned in equation (10) . K . K . K . K . kJ function de?ned in equation (69) function de?ned in equation (11) sequential variable Z3 function de?ned in equation (12) n P Pin q Q1 Q2 R Rm Rmax Rmin t sequential variable revenue per cycle . . . . . . . . . . . . . . . power input . . . . . . . . . . . . . . . . . . speci?c heat transfer . . . . . . . . . . . . . heat ?ow from refrigerator to heat sink . . . heat ?ow from heat source to cold refrigerant cooling load . . . . . . . . . . . . . . . . . . cooling load at maximum pro?t . . . . . . . maximum cooling load . . . . . . . . . . . . minimum cooling load . . . . . . . . . . . . time . . . . . . . . . . . . . . . . . . . . . . $ ·s-1 . . kW . . kW . . kJ . . kJ . . kW . . kW . . kW . . kW . . s Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 function de?ned in equation (17) function de?ned in equation (49) function de?ned in equation (50) function de?ned in equation (51) function de?ned in equation (52) function de?ned in equation (60) function de?ned in equation (61) function de?ned in equation (62) function de?ned in equation (64) function de?ned in equation (65) function de?ned in equation (66) . K . K . K . K . K . K . K . kW . kW . kW Greek letter * Correspondence and reprints. E-mail address: [email protected] (C. Wu). ? 2001 Éditions scienti?ques et médicales Elsevier SAS. All rights reserved S1164-0235(01)00031-0/FLA heat conductance . . . . . . . . . . . . . . . heat conductance . . . . . . . . . . . . . . . kW·K-1 kW·K-1

? ?m L. Chen et al. / Exergy Int. J. 1(4) (2001) 295–302 temperature index de?ned in equation (33) temperature index at the condition of maximum pro?t d function de?ned in equation (12) e COP ec Carnot COP em COP at maximum pro?t, ?nite-time exergoeconomic COP bound R COP at maximum cooling load ? Carnot coef?cient s rate of entropy production . . . . . . . . . . kW·K-1 S change in entropy . . . . . . . . . . . . . . . kJ·K-1 t total time . . . . . . . . . . . . . . . . . . . . . . . . . s fA price of exergy . . . . . . . . . . . . . . . . . . $ ·kJ-1 fW price of work . . . . . . . . . . . . . . . . . . $ ·kJ-1 ? pro?t ratio . . . . . . . . . . . . . . . . . . . . . $ ·s-1 ?m optimal pro?t . . . . . . . . . . . . . . . . . . . $ ·s-1 ?max maximum pro?t . . . . . . . . . . . . . . . . . . $ ·s-1 1. INTRODUCTION The Carnot engine proposed in 1824 operates on re- versible process principles. As a consequence, this hy- pothetical engine produces the maximum possible work for a given heat source and sink temperatures, but gener- ates zero power because it has to operate at an in?nitely slow pace. Its thermodynamic ef?ciency, which has long been used as the standard against which all real engine ef?ciencies are measured, is unrealistically high. In 1975 did Curzon and Ahlborn [1] pioneered an analysis that accounts for the irreversibilities of ?nite-time heat trans- fer to and from the engine. Such an endoreversible en- gine can generate useful power. Because of external ir- reversibil