Efect of heat transfer law on the finite-time exergoeconomic performance of a Carnot refrigerator (página 2)

Partes: 1, 2

ities, its ef?ciency at maximum power, which is termed the “?nite-time thermodynamic ef?ciency”, is less than that of the Carnot ef?ciency. Since ?nite-time thermodynamics was ?rst advanced in 1975, many au- thors have studied the effect of irreversibilities on the per- formance of thermodynamic processes and cycles. Some detailed literature surveys of ?nite-time thermodynamics were given by Sieniutycz and Salamon [2] and Chen et al. [3, 4]. Some authors [5–18, including] have assessed the effect of ?nite-rates of heat transfer on the performance of irreversible refrigerators. The objective functions in ?nite-time thermodynam- ics are often pure thermodynamic parameters includ- ing power, ef?ciency, entropy production, effectiveness, cooling load, speci?c cooling load, COP and loss of ex- ergy. Salamon and Nitzan [19] viewed the operation of the endoreversible heat engine as a production process 296 with work as its output. They carried out the economic optimization of the heat engine with the maximum pro?t as the objective function [20]. A relatively new method that combines exergy with conventional concepts from long-run engineering eco- nomic optimization to evaluate and optimize the de- sign and performance of energy systems is exergoeco- nomic (or thermoeconomic) analysis. Some detailed lit- erature surveys of the exergoeconomics were given by Sieniutycz and Salamon [2] and Tsatsaronis [21]. Sala- mon and Nitzan’s work [19] combined the endoreversible model with exergoeconomic analysis. We termed it as ?nite-time exergoeconomic analysis [22, 23] to distin- guish it from the endoreversible analysis with pure ther- modynamic objectives and the exergoeconomic analy- sis with long-run economic optimization. Similarly, we termed the performance bound at maximum pro?t as ?nite-time exergoeconomic performance bound to distin- guish it from the ?nite-time thermodynamic performance bound at maximum thermodynamic output. Based on the work for heat engines [19, 22, 23], such a method has been extended to Newton’s Law two-heat-reservoir re- frigerator [10, 11] and heat pump [24], and the three-heat- reservoir refrigerator [25] and heat pump by Chen et al. [26]. Heat transfer affects the performance of endoreversible cycles. A few authors [7, 9, 13, including] have assessed the effect of heat transfer laws on the cooling load versus COP characteristics for a refrigerator. A new step taken in this paper is the estimation of the pro?t versus the COP characteristics and analysis of the ?nite-time exergoeco- nomic performance based on a relatively general heat transfer law, q ? (T n ), where n is a heat transfer expo- nent. The heat transfers obey Newton’s Law when n = 1, the linear phenomenological law in irreversible thermo- dynamics when n = -1, and the radiative heat transfer law when n = 4. 2. ANALYSIS 2.1. The relation between optimal pro?t and cop An endoreversible Carnot refrigerator is shown in ?gure 1. The only irreversible processes in the cycle are the two heat transfer processes from the refrigerator to the heat sink and from the heat source to the refrigerator. To analyze this cycle, we assume that the temperatures of the heat sink, heat source, warm refrigerant in the heat

edu.red -1 n n n -1 n n where n n n n -1 (10) (11) (12) (n+1)/2 n n (n-1)/2 n (13) (n-1)/2 n n (n-1)/2 n (14) are (1-n)/2 2 n n n (4) L. Chen et al. / Exergy Int. J. 1(4) (2001) 295–302 output of the refrigerator (A) is: A = Q2 (T0 /TL – 1) – Q1 (T0 /TH – 1) = Q2 ?2 – Q1 ?1 (6) where ?i is the Carnot coef?cient of the reservoir i. The pro?t (?) is calculated for the cycle period as follows. If fA is the value price of exergy output, we have a revenue function (P ) per cycle: P = fA A/t (7) We assume that the only input to the production process is the work input (W ) taken from the motor. This corresponds to a cost per unit time (C): C = fW W/t (8) Figure 1. Endoreversible Carnot refrigerator. rejection process, and cold refrigerant in the heat addition process are TH , TL , TWH , and TWL , respectively. Thus heat ?ows from the heat source to the cold refrigerant across a temperature difference of (TL – TWL ) and heat ?ows from the warm refrigerant to the heat sink across a temperature difference of (TWH – TH ). Assuming the heat transfers between the refrigerant and the reservoirs obey a generalized heat transfer law, q ? (T n ), then Q1 = a TWH – TH t1 (1) Q2 = ß TL – TWL t2 (2) where Q1 and Q2 are heat ?ows from the refrigerator where fW is the price of work. Using equations (1)–(8), the pro?t of the refrigerator is obtained ? = P – C = (Q2 ?2 – Q1 ?1 )fA – (Q1 – Q2 )fW /t = afA (Z2 – Z1 Z3 ) / Z3 TWL Z3 – TH + d2 TL – TWL (9) Z1 = ?1 + fW /fA Z2 = ?2 + fW /NA Z3 = 1 + e to the heat sink and from the heat source to the cold refrigerant. Constants a and ß are the heat conductances (product of heat transfer coef?cient and heat transfer and d = (a/ß)0.5 surface area) between warm refrigerant and heat sink and between heat source and cold refrigerant. Variables t1 and t2 are the times required to transfer an amount Q1 and Q2 Taking the derivative of ? with respect to TWL and setting it equal to zero (??/?TWL = 0) gives of heat, respectively. Neglecting the time required for the two isentropic TWL,opt = dTH /Z3 + TL / 1 + dZ3 processes, the total time (t ) required for the whole cycle The corresponding warm refrigerant temperature is: is: t = t1 + t2 (3) The COP (e) and the work input (W ) to the refrigerator TWH,opt = dTH Z3 + Z3 TL / 1 + dZ3 Substituting equation (13) into equation (9) yields: e = (Q1 /Q2 – 1)-1 = (TWH /TWL – 1)-1 ?m = afA TL – TH /Z3 (Z2 – Z1 Z3 )/ d + Z3 W = Q1 – Q2 (5) (15) Assuming the environment temperature is T0 and the following relation holds: T0 TH > TL , the exergy Equation (15) is the main result of this paper. It deter- mines the optimal pro?t for the given COP and the opti- 297

edu.red (3n+1)/2 n n+1 n n n (n+1)/2 (n-1)/2 n n n -1 (1-n)/2 2 n n n n n n 2 (20) n n n 0.5 (1-n)/2 2 L. Chen et al. / Exergy Int. J. 1(4) (2001) 295–302 mal COP for the given pro?t. It is called the ?nite-time exergoeconomic fundamental optimal relation or optimal pro?t versus COP characteristics. Equation (15) indicates that pro?t is zero when e = c = (TH /TL – 1)-1 and e = (Z2 /Z1 – 1)-1 . Hence, there exists an extreme pro?t for the refrigerator. The maximum pro?t may be found by taking the derivative of ?m with respect to e and setting it equal to zero (??m/?e = 0). For maximum pro?t, the COP bound (em ) satis?es the equation dZ1 TL Z4 + nZ1 TL Z4 – (n – 1)Z2 TL Z4 + (n – 1)dZ1TH Z4 – ndZ2 TH Z4 where S is the change in entropy over the cycle. Comparing equations (21) and (20) gives: ?m = -fA T0 s = -fA S/t 0 (22) That is, the pro?t maximization approaches the rate of entropy production minimization,or in other words, the minimum waste of exergy (T0 S). Equation (22) indicates that the refrigerator is not pro?table regardness of the COP at which the refrigerator is operating. Only if the refrigerator is operating reversibly (e = eC ) will the revenue equal the cost, and then the maximum pro?t will equal zero. (The corresponding rate of entropy production is also zero.) – Z2 TH = 0 (16) Therefore, for any intermediate (fW /fA ), the ?nite- where Z4 = 1 + em (17) time exergoeconomic performance bound (em ) lies be- tween the ?nite-time thermodynamic performance bound and the reversible performance bound. em is related to the The COP ( m ) is different from both the classical re- versible COP bound (eC ) and the ?nite-time thermody- namic COP bound (COP at the maximum cooling load, R ), and was termed as ?nite-time exergoeconomic COP bound. It is dependent on TH , TL , T0 , d, n, and (fW /fA ). Note that for the process to be potentially pro?table, the following relationship must exist: 0 < (fW /fA ) < 1, because one unit of work can give rise to at least one unit of exergy output. As the price of exergy becomes very large compared with the price of work, i.e., fW fA , (fW /fA ) ? 0, and T0 = TH , equation (15) becomes: ?m = fA ?2 R (18) where R is the optimal cooling load for the given COP [9, 13]. R = Q2 /t = a TL – TH /Z3 / d + Z3 (19) That is, the pro?t maximization approaches cooling load maximization. latter two through the price ratio, and the associated COP bounds are the upper and lower limits of em . 2.2. Optimal pro?t versus cop characteristics for three common heat transfer laws The optimal pro?t versus COP characteristics is dis- cussed in this section for three common heat transfer laws: Newton’s Law (n = 1), the linear phenomenologi- cal law in irreversible thermodynamics (n = -1), and the radiative heat transfer law (n = 4). Case n = 1 In this case, equation (15) becomes: ?m = afA (TL – TH /Z3 )(Z2 – Z1 Z3 )/(1 + d)2 (23) The solution of equation (16) is: On the other hand, as the price of exergy approaches the price of work, i.e., (fW /fA ) ? 1, equation (15) em = TH Z2 /(TL Z1 ) 0.5 – 1 -1 (24) becomes: ?m = -fA aT0 TL – TH /Z3 (TL Z3 – TH )/ (1-n)/2 2 TH TL d + Z3 The rate of entropy production of the refrigerator for the given COP is s = S/t = R(Z3 /TH – 1/TL) = a TL – TH /Z3 (TL Z3 – TH ) The maximum pro?t is: ?max = afA (TH Z2 )0.5 – (TL Z1 )0.5 /(1 + d)2 (25) The corresponding cooling load is: Rm = aTL 1 – TH Z1 /(TL Z2 ) /(1 + d)2 (26) As T0 = TH and (fW /fA ) ? 0, then equation (23) becomes: / TH TL d + Z3 (21) ?m = fA ?2 R (27) 298

edu.red 2 and (30) or (32) 1-? ? 1-? ? / , 2 (43) -1 -1 (37) L. Chen et al. / Exergy Int. J. 1(4) (2001) 295–302 where R = a(TL – TH /Z3 )/(1 + d) (28) The cooling load (R) is a monotonic decreasing function of e, and R approaches Rmax = aTL /(1 + d)2 as e approaches emin = 0 and R approaches Rmin = 0 as e approaches ec . An interesting question is how to determine the pro?t for a given cooling load R or a given power input, Pin = W/t . From equation (28), Equation (37) indicates that ? = 1 and ? = ?m when ? = ?A ?A = (1 + d)-1 (38) and ?m = ?max when ? = ?m . ?m = 1 + ln(Z2 /Z1 )/ ln(TH /TL ) /2 (39) Equations (35)–(39) are termed the pro?t and COP holographic spectra of Newton’s Law for refrigeration systems. From the view of compromise optimization of pro?t and COP, we have e = (TH – TL + DPin )2 + 4TL DPin 0.5 1

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