Energy, Exergy, and Second Law Performance Criteria

ARTICLE IN PRESS

Energy 32 (2007) 281–296 www.elsevier.com/locate/energy

Energy, exergy, and Second Law performance criteria

Noam Liora,Ã, Na Zhangb

aDepartment of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA bInstitute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100080, PR China

Received 14 October 2005

Abstract

Performance criteria, such as efficiencies and coefficients of performance, for energy systems, are commonly used but often without sufficient understanding and consistence. The situation becomes particularly incoherent when simultaneous energy interactions of different types, such as work, heating and cooling, take place with a system. Also, the distinction between exergy and Second Law efficiencies is not clearly recognized by many. It is attempted here to clarify the definitions and use of energy and exergy based performance criteria, and of the Second Law efficiency, with an aim at the advancement of international standardization of these important concepts. r 2006 Elsevier Ltd. All rights reserved.

Keywords: Thermodynamics; Efﬁciency; Exergy; Second Law; Performance

1. Introduction easily converted to energy cost efficiencies if the prices of the energy forms of the useful outputs and paid inputs are There are many ways to assess energy system perfor- known. Here we define as ‘‘useful’’ all energy interactions mance, and they must be adapted to the particular use they that have been used by the system ‘‘owner’’, usually in are put to. While the underlying concepts of such terms of monetary value, and ‘‘paid’’ all energy interac- assessments are often well known and documented for a tions that have a direct cost, usually monetary, to the number of systems and cases, there are no clear agreements system owner. Thus, for example, the heat inputs that come or rules about efficiency definitions, and authors often use from the environment for which the owner does not need to different, and sometimes unsuitable, efficiency definitions pay are not included. We hasten to add that analyses that for the same systems. The situation becomes particularly address environmental impact will include exchanges with incoherent when simultaneous energy interactions of the environment too. different types, such as work, heating and cooling, take Since such energy-based criteria do not account for the place with a system. This prevents logical comparison of quality of energy, expressed as exergy, exergy-based criteria results at best, and wrong results at worst. This paper is are also appropriate as they account better for use of intended to provide some clarifications and uniformity in energy resources and give much better guidance for system that area, make a few proposals, and start a discussion that improvement. They also can be converted to exergy cost would hopefully lead to accelerated international standar- efficiencies if the exergy values of the useful outputs and dization of these important concepts. paid inputs can be rationally priced. The most common energy system performance assess- Another set of criteria assess the difference between the ment criteria are energy based (‘‘first law’’) and they are performance of a system relative to an ideal one (reversible) useful for assessing the efficiency of energy use, and can be that operates between the same thermodynamic limits. We shall call these Second Law-based criteria, although many authors use this term for exergy-based criteria. ÃCorresponding author. Tel.: +1 215 573 6000; fax: +1 215 573 6334. Ultimately, decisions on best designs are most often E-mail address: [email protected] (N. Lior). based on economical considerations, in which energy (or

Nomenclature ZIem embodied energy efficiency ZIu utilitarian (task) energy efficiency a specific exergy, J/kg; price of energy unit, $/MJ ZI$ economic energy efficiency A exergy, J ZII Second Law efficiency b price of exergy unit, $/MJ ZQ useful heat production efficiency in a cogenera- cp specific heat at constant pressure, J/kg k tion cycle, Eq. (92) cv specific heat at constant volume, J/kg k Zw work production efficiency in a cogeneration COP coefficient of performance cycle, Eq. (91) ~ COP reversible cycle coefficient of performance, Eq. ZÎ energy efficiency when the total energy content (105) of the system is taken into consideration E energy, J Z~hp heat input based reversible cycle efficiency h specific enthalpy, kJ/kg definition, Eq. (104) I irreversibility, J Z~hu heat output based reversible cycle efficiency j the number of useful work or inputs and definition, Eq. (106) outputs tpayback energy payback time, years ke specific kinetic energy, J/kg m_ mass flow rate of a stream Subscripts n the number of paid-for heat inputs q heat per unit mass, J/kg 0 dead state pe specific potential energy, J/kg a absorber P pressure, Pa c cooling, or cold temperature reservoir Q heat, J con condenser _ Q heat invested in plant construction, J cu for refrigeration use rc the useful cooling to work output ratio, Eq. cp paid-for refrigeration (89) d direct rh the useful heat to work output ratio, Eq. (90) e environment of a system R universal gas constant em embodied, Eq. (12) RI,e First Law environmental impact ratio, Eq. (13) E direct energy invested in plant construction s specific entropy, J/kg k f heating or cooling fluid S entropy, J/K h heating, or high temperature reservoir Sgen entropy generation, J/K hp paid-for heat t time, s hu for heating use T temperature, K i incoming T^ entropic temperature, Eq. (54), K I First Law v specific volume, m3 II Second Law w specific work, J/kg j index, the jth heat or work exchange W work, J k number of material products of a plant W^ work invested in plant construction, J l labor x price per unit exergy of heat, $/MJ L index of energy needed for restoration of the environment to its original state Greek letters m materials o outgoing e exergy efficiency p paid for ^t The ‘‘total’’ (or ‘‘overall’’) unsteady state rev reversible exergy efficiency s system ~t The ‘‘total’’ (or ‘‘overall’’) steady state exergy t total efficiency, including the system exergy u useful ZI energy (First Law) efficiency

exergy) are only one part, and sometimes not the most energy conversion device, they can be applied at any level, signiﬁcant one, where the other parts include capital such as to different components, inside spatial and investment, labor, insurance, taxes, etc. temporal processes, and down to the smallest particle It is also worth noting that while performance criteria interactions, when there is an interest in that kind of are most often applied to the entire system, such as plant or exploration. ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 283

2. Energy-based criteria As usual, both the work and heat terms may be composed of flow and direct components, or in general 2.1. Introduction W ¼ mke_ ðÞþþ pe W d ; Q ¼ mh_ þ Qd , (3) There are many definitions of energy efficiency, based on where m_ is the mass flow rate of the stream carrying the need (or greedy), and some of the more common ones are specific kinetic and potential energy work, or h in or out of reviewed below. the system, and Wd and Qd are the direct energy input to- and output from- the system. For work terms the latter may be a direct mechanical interaction with the system, 2.2. Total energy efficiency such as a stirrer inserted into it, and for heat ones it may be irradiation or any exothermic reaction. We start with the energy conservation equation for a A total energy efficiency ZÎt can be defined in its broadest system (Fig. 1) undergoing a process from state 1 to state 2, manner as P P where the values of the system energy at these states are E1 E2 þ jW o;j þ jQo;j and E , respectively, and which undergoes j work and heat Z^ ¼ P P 2 It E þ Q þ W interactions 1 j i;jj i;j !E2 þ W o þ Q Q þ Q X X X X ¼ o;h o;c o;e ¼ 1. ð4Þ E1 þ W i þ Qi;h Qi;c þ Qi;e E1 E2 ¼ W o;j þ Qo;j Qi;j þ W i;j j j j j Obviously, the ratio of all energy outputs to inputs, Eq. ¼ W þ Q Q þ Q (4), is always equal to 1 and is thus useless for system oo;h o;c o;e performance assessment, although it should always be used W i þ Q Q þ Q , ð1Þ i;h i;c i;e in computations as a part of results validation. where in the second line we use the notation Most often, the steady-state total energy efficiency is X X defined without consideration of the energy value of the W o W o;j W i W i;j, system, E, to more clearly focus on the energy inputs and j X j outputs only, and thus the best known form of the steady- Qo;h Qo;c þ Qo;e Qo;j, state total energy efficiency takes the form j P P X W o;j þ Q W o þ Q Q þ Q Z Pj P j o;j ¼ o;h o;c o;e ¼ 1. Qi;h Qi;c þ Qi;e Qi;j ð2Þ It Q þ W i;j W i þ Q Q þ Q j j i;j j i;h i;c i;e (5) for compactness; the terms Q express the thermal energy interactions of heating, cooling, and heat interactions with 2.3. Common energy efficiencies the environment. There are variants of the total energy efficiency that W Q selectively include some of the inputs and/or outputs to analyze some particular aspect of energy use, and examples

State State2 are shown below. E1 1 E2 2.3.1. The common energy efﬁciency Z Environment Iu The common (utilitarian, or ‘‘task’’) energy (First Law)

Qo,e Qo,c efﬁciency ZIu is based on the practical energy needs (work, heat, refrigeration) of the system owner that are satisﬁed by

To,c Wo the system, divided by the energy inputs to the system that To,e Qo,h must be ‘‘paid’’ for. The practical extension of this • thermodynamic definition is to define the numerator terms m o as having a monetary benefit value, and those in the To,h denominator as having a monetary cost, to the system E owner. For a general system with used work, heat and T i,c refrigeration outputs, and paid work, heat and refrigeration • mi inputs (a preliminary discussion of such combined system Ti,h Ti,e efficiencies can be found in [1,2]), it is in general usually Qi,c W best expressed in terms of their absolute values as i P P P P Qi,e Q W W þ Q þ Q i,h j u;j Pj p;j Pj hu;j j cu;j ZIu ¼ , (6) Fig. 1. Energy interactions. j Qhp;j þ j Qcp;j ARTICLE IN PRESS 284 N. Lior, N. Zhang / Energy 32 (2007) 281–296 noting that in this caseP it is assumed P that positive net power Table 1 is produced for use, W 4 W , and thus the paid Energy payback for some power generation systems j u;j P j p;j work inputs (consumption) W are subtracted from P j p;j Type of system Energy Reference payback, years the useful work outputs j W u;j in the numerator of Eq. (6) rather than added to the denominator. This is an Coal power plant 3.6 [3] important distinction, because the latter definition is more Nuclear power plant 2.6 [3] consistent with thermodynamic logic but gives higher Solar domestic hot water system 1.3–2.3 [4,5] values of Z that favor the system vendor (see Appendix Solar photovoltaic system 2–6 [5,6] 1u Wind-electric generator 0.25–0.67 [7] A for the somewhat trivial proof). P P If j W u;j o j W p;j , the definition of ZIu that is consistent with Eq. (6), favoring the customer, is P P where the numerator is the total energy (the terms marked j Qhu;j þ j Qcu;j Z ¼ P P P P . (7) by the overbar ^) invested in the direct energy demand Iu Q þ Q þ W W j hp;j j cp;j j p;j j u;j (subscript E), and energy content of the materials (m), and labor (l) needed for the system construction, and the 2.3.2. The coefficient of performance, COP denominator is the annual production of useful energy (net This terminology is used mostly in evaluating the work, heat, refrigeration). performance of systems that produce primarily cooling, In fact, a more consistent definition of tpayback would and the definition is actually identical to that of ZIu , Eq. also include in the numerator of Eq. (10) the total energy (7). When the only product is cooling, and no cooling investment, embodied in the fuel, materials and labor inputs to the system are present, Eq. (7) is reduced to needed for the plant operation over its lifetime (L years), P and would also included in the denominator the energy Q P Pj cu;j P embodied in non-energy k products of the plant, ZIu ¼ ¼ COP, (8) P Q þ W p j W u j j hp;j j ; j ; kjjEk m. To create a more equitable comparison between systems which is commonly used to characterize cooling systems that produce different impacts on the environment, these that have both heat and work inputs, such absorption ones. embodied energies should also include the energy needed For cooling systems that do not use heat inputs, Eq. (8) is for restoration of the environment to its original state (the reduced to second term in the numerator, index L). With the annual P Q embodied heating and cooling investments (the second P j cuP;j COP ¼ . (9) term in the numerator), the equation would then become j W p;j j W u;j

In fact, fewP cooling systems produce useful work and then tpayback;t the term W in Eqs. (8) and (9) is zero. P P P P P P j u;j ^ ^ ^ L j W j þ j Qh;j þ j Qc;j þ 1 j Qhp;j þ j Qcp;j E;m;l E;m:l ¼ hi . P P P P P j W u;j j W p;j þ j Qhu;j þ j Qcu;j þ kjjEk 2.4. Embodied energy efficiency m peryear ð11Þ It is often of interest (or should bey) to include in the energy efficiency assessment also the energy embodied in the In energy efficiency form, such a more comprehensive production of the plant, in the materials produced by it, and expression would be an extension of Eq. (6), which adds in the materials and labor needed for its operation and for the annualized investment in the construction of the plant the distribution of its material products to the customer. and in the materials and labor needed in its regular Such a criterion answers, for example, the commonly posed operation question on the length of time that it takes for an energy conversion system to generate the energy originally required ZI;em hi for its manufacturing and operation. For example, Table 1 P P P P P W u;j W p;j þ Q þ Q þ jjEk j j j hu;j j cu;j k m shows the numbers of years of successful operation that it ¼ , P P P P P j W j þ j Qh;j þ j Qc;j þ j Qhp;j þ j Qcp;j takes to pay back the energy invested in the construction of E;m;l E;m;l several types of energy conversion systems. ð12Þ The equation commonly used for that energy payback P period tpayback , in years, is where the last term in the numerator, jjE , denotes k k m P P P ^ ^ ^ the embodied energy in the k types of materials produced j W j þ j Qh;j þ j Qc;j E;m;l by the plant, and the first (bracketed) term in the t ¼ P P P P , payback denominator is composed of the annualized energy j W u;j j W p;j þ j Qhu;j þ j Qcu;j peryear investment in the production of the plant, and the second (10) term is the annual embodied energy invested in the energy ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 285 sources, materials and labor needed for its regular 3. Exergy efficiency, e operation. All the energy terms are rates. 3.1. Introduction 2.5. Energy criteria considering environmental effects Just as there are various definitions of energy efficiencies as discussed above, there are also various definitions of In some circumstances the objective of the analysis may exergy efficiency. Sorting out the different definitions to include the energy impact of the system on the environ- avoid misunderstandings and contribute to uniformity has ment, including heat rejection to the environment or heat been the subject of a number of papers from Germany in absorption from it, even though no direct (or immediate) the 1950s and 1960s (cf. [9–14]). We discuss here overall monetary cost or benefit may be associated with it. This efficiency, the utilitarian (or ‘‘task’’) efficiency, and some also would be of interest for finding the effects on ambient ways to define the exergy. cooling or heating water consumption, etc. In that case we Of help in these definitions is the exergy accounting of can define a thermal environmental impact ratio, R , all of I,e the system. For a thermodynamic system going through the heat interactions with the environment (heat rejection interactions between an initial state 1 and a later state 2, and absorption) to the sum of the useful net power, heating where the system exergy values are A and A , and cooling produced, as s,1 s,2 P respectively, the equation representing this accounting is Q X X R ¼ P P i Pe;i P . (13) A A ¼ A A I, (16) I;e s;2 s;1 i i o o i W u;i i W p;i þ i Qhu;i þ i Qcu;i where Ai and Ao are the exergy inputs and outputs to and from the system, respectively, and I is the total exergy loss (or destruction). Due to the Second Law, I X 0. 2.6. Evaluation criteria based on economics In steady state As,1 ¼ As,2, and Eq. (16) gives X X A ¼ A I. (17) The focus of this paper is on the technical aspects of o o i i energy conversion system performance criteria, but for The irreversibility I is linked to the entropy generation Sgen completeness it is noted that decisions on industrial/ by the equation commercial systems are typically based on economic I ¼ T S , (18) considerations. As stated in the Introduction, the energy 0 gen performance, as characterized by the different performance where T0 is the dead state temperature (in absolute units). criteria described above, is only one of the components of the total cost. Typical energy performance criteria include 3.2. The ‘‘total’’ (or ‘‘overall’’) exergy efficiency, ^t payback period, return on investment (ROI), and life cycle cost analysis (cf. [8]), and those can be either restricted to P ð Þþ the system exergy at state 2 oall exergy outputs the direct energy streams, or to the energy streams included ^t P ðthe system exergy at state 1Þþ all exergy inputs embodied energy, for example as in Eq. (12). Nevertheless, P i we can, for some specific purposes, define a First Law A þ A ¼ s;2 Po o ð19Þ economics efficiency, ZI$, by including the prices of the As;1 þ iAi different heat and work inputs and outputs, and Eq. (6) and, as before, the ingoing and outgoing exergy quantities becomes may include work, heat and cooling, with or without mass hiP P P P flows. Incorporating Eq. (16) into (19) gives jbu;j W u;j jbp;j W p;j þ jahu;j Qhu;j þ jacu;j Qcu;j Z ¼ P P P I$ þ A þ A I I jahp;j Qhp;j jacp;j Qcp;j s;1 Pi i P ^t ¼ ¼ 1 . (20) (14) As;1 þ iAi As;1 þ iAi and Eq. (7) becomes This also shows that ^t p 1, where the equality applies to P P completely reversible processes. This is unlike the overall ahu;j Q þ acu;j Q Z ¼ hi j hu;j j cu;j , energy efficiency ZIt [Eqs. (4) and (5)] which is always equal I$ P P P P jahp;j Qhp;j þ jacp;j Qcp;j þ jbp;j W p;j jbu;j W u;j to unity, because energy is conserved, while exergy always decreases in all real (irreversible) processes. (15) ^t is used when the overall exergy efficiency of a system where the coefficients a and b are the specific prices of the needs to be calculated, without interest in some of the different types of heat and work, respectively, say in $/MJ . individual outputs of the system. It is especially applicable Eqs. (14) and (15) express how much money is gained from when the system input and output streams do not maintain the useful heat and work products per monetary unit their integrity and the major part of the output is useful. It invested in the paid heat and power consumed by the also has a more thermodynamic flavor than other efficiency system. definitions, in that it evaluates the efficiency of a process ARTICLE IN PRESS 286 N. Lior, N. Zhang / Energy 32 (2007) 281–296 giving equal consideration to all outputs and inputs say In $/MJ. Eqs. (26) and (27) express on an exergy basis regardless of whether they are being used or paid for. how much money is gained from the useful heat and work For example, it offers a good starting point for ecological products per monetary unit invested in the paid heat and analysis of processes since it takes into equal consideration power consumed by the system. exergy components as diverse as useful work, heat and product materials on the one hand, and material and 3.4. Exergy definitions heat emissions to the environment on the other, on the output side, and paid fuel, work, and materials on the one As opposed to energy-based performance criteria, where hand, and ‘‘free’’ environmental heat on the other, on the the definitions of the energy interactions with the system input side. are fairly straightforward, it is not so for exergy. To start, In steady state As,1 ¼ As,2 ¼ As and Eqs. (19) and (20) any work (including electrical energy) interactions are still become straightforward, because they are pure exergy and thus the P þ exergy values of such interactions are equal to their energy 1 oAo As ~t P (21) values. Non-work energy interactions (heat for example) 1 þ Ai As have values of exergy that depend on both the type of the i energy interaction and the definition of the control system and under consideration. The latter depends on the objective of the performance analysis. We can classify the types of non- IP ~t ¼ 1 , (22) work energy interactions and of typical objectives as A þ A s i i follows. respectively. Most often, the steady-state total exergy efficiency is 3.4.1. Straight heat input or output defined without consideration of the exergy value of the In general, the exergy of the system heat interactions of a system, As, to more clearly focus on the exergy inputs and system at constant temperature T with an environment at outputs only, and thus the best known form of the steady- the dead state temperature T0 is calculated from state total exergy takes the form P T A ¼ Q ð1 0Þ, (28) oAo I h;i h;i t P ¼ 1 P . (23) T iAi iAi when T4T0 and Qh,i is the heat output from the system to the environment (or to a heat load at T0), and T 3.3. The task (or ‘‘utilitarian’’) exergy efficiency, e A ¼ Q ð 0 1Þ, (29) u c;i c;i T when T T and Q is the heat input from the cooling load Using the same logic as in definition of ZI, eu can be o 0 c,i defined as the ratio between the exergy outputs useful to at T to the system. So in these cases the task exergy the owner, and the exergy inputs paid by the owner. The efficiency for a system with just heat input (‘‘paid’’, Qhp,i)at most common is the steady-state definition, and thus the temperature T and work output (used, Wu) is defined as equivalent of Eqs. (6), (7), (14), and (15) are Eqs. (24) and W u (25) below: u ¼ (30) Qhp;i½1 ðT0=TÞ P P P P W u;j W p;j þ Ahu;j þ Acu;j j Pj Pj j and for a system with just work input (‘‘paid’’, Wp) and u ¼ , (24) j Ahp;j þ j Acp;j refrigeration heat input at temperature T (used, Qcu,i) P P Qcu;i½ðT 0=TÞ1 A þ A ¼ . (31) P Pj hu;j Pj cu;j P u u ¼ , (25) W p j Ahp;j þ j Acp;j þ j W p;j j W u;j Eqs. (28) and (29) are often used in defining the exergy hi P P P P value of heat, but it should be clarified that neither of these bu;j W u;j bp;j W p;j þ xhu;j Ahu;j þ xcu;j Acu;j j Pj Pj j temperatures, nor the heat quantities, are necessarily $ ¼ , jxhp;j Ahp;j þ jxcp;j Acp;j constant during the process in the system under considera- (26) tion. The heat sources may be of variable temperature, and the heat input rate and even the dead state may vary during P P jxhu;j Ahu;j þ jxcu;j Acu;j the process also. In that case, the more general expression ¼ hiP P P P , $ of Eq. (28) for example, for a process starting at time t and jxhp;j Ahp;j þ jxcp;j Acp;j þ jbp;j W p;j jbu;j W u;j 1 ending at time t2 would be (27) Z t2 T where the coefficients x and b are the specific exergy value A^ ¼ Q ð1 0Þ dt, (32) h;i h;i T prices of the different types of heat and work, respectively, t1 ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 287 where all the parameters in the integral may vary with time. Table 2 For the efficiency, Eq. (30) becomes External stream-carried exergy configurations and magnitudes Z t2 W ^ ¼ u dt. (33) u Q ð1 ðT =TÞÞ t1 hp;i 0 Similar integrations would also be used in Eqs. (29)–(31) for the refrigeration system when the integrand varies with time.

3.4.2. Stream-carried exergy Exergy can be carried in and out of systems with a mass flow stream at the rate ma_ , where m_ is the mass flow rate and a is the specific exergy. a should include all the relevant exergy components including kinetic, potential, thermal, strain, and chemical as explained in detail in [9]. The sketch Table 3 in Table 2 shows two common flow configurations: (a) in External stream-carried exergy configurations and magnitudes column 2 depicts a case in which the stream does not mix with the system and maintains its integrity, and (b) in column 3 depicts a case where a stream enters the systems, loses its integrity in it, and another stream exits the system. From a strict thermodynamic perspective of exergy analysis, the exergy inputs should be calculated using Eqs. (34) and (35) given in Table 2, but it is noteworthy that in some cases it may be decided that either the inflow or outflow exergy streams have no value for the purposes of the analysis. In these cases it is assumed that a1 and/or a2 are equal to zero. For example, this would be the case when a stream of fluid is used to supply heat to a system and exits the system at conditions which are deemed, often for economical reasons, unusable for further energy extraction. streams that are not directly connected, but the one (b) in Another example would be when a stream is cooled by the column 3 of Table 3 includes the interactions between these system to a temperature much lower than its entrance one, streams and the system, with the associated irreversibility so that the outlet exergy of the stream is much lower than in Eq. (37) of Table 3. A sample case is a combustor with its entrance one, and thus the outlet exergy may be inputs of fuel and oxidant, and an output of reaction considered negligible. products, when the exergy losses due to the combustion The configurations described in Table 2 ignore exergy process (including mixing, chemical reaction, heat transfer) losses during the exchange between the stream and the are wished to be taken into account (cf. [15]). Use of such a system, in other words it is implicitly assumed that these model is the basis for ‘‘intrinsic’’ exergy analysis that losses are either negligible (infinite exchange coefficients) or allows detailed (even spatial and temporal) determination that the analysis is not aimed at this purpose. A more of exergy changes in all interaction processes (cf. [15–18]). rigorous model, which includes the exchange exergy losses, is described in Table 3. (a) in column 2 describes a system 4. Second Law efficiency where the external stream 1e-2e and a system stream 1s- 2s are interacting while maintaining their integrities. A 4.1. Introduction sample case is when stream 1s-2s is the evaporator in a refrigeration system, and stream 1e-2e is the fluid that is Given the objectives of the system under consideration, being cooled by the evaporator (refrigerator). such as power production, cooling, heating, materials Eq. (36) in Table 3 represents the exergy balance in this processing, etc., and the thermodynamic parameter limits, case, where Ie2s is the irreversibility due to the interaction such as the top and bottom temperatures, pressures, and between the streams, and allows the inclusion of this other thermodynamic potentials, it is useful to compare the irreversibility in the system efficiency calculation by performance of the system with one that would deliver the ascribing the exergy (input or output) value to the external, best thermodynamic performance between the same limits, rather than the system, stream, as was done in the model i.e. one in which all the processes are reversible. This (a) of Table 2, column 2. indicates clearly a potential for improvement. Similar to the model (b) of Table 2, column 3, that of the The Second Law efficiency is defined here ([2] and many model (b) of Table 3, column 3 represents input and output others) as the ratio of the First Law efficiency of the system ARTICLE IN PRESS 288 N. Lior, N. Zhang / Energy 32 (2007) 281–296 to the efficiency of a reversible system operating between (P1, T1) and state 2s (P2s T1s) that is on the same isobar as the same thermodynamic states, state 2 but has the same entropy as state 1. The premise of Z this definition differs from the definition of the Second Law Z ¼ I (38) II Z efficiency in a very important way: the end states for the rev real and reversible processes in this definition, Eq. (39), are It is noteworthy that the thermodynamic states must be different, while for the definition of the Second Law clearly and correctly defined, especially in view of some efficiency in Eq. (38) the end states for the real and practices entrenched in engineering thermodynamics that reversible processes are the same. Eq. (38)-based definition may be misleading. A good and simple example for such of the turbine isentropic efficiency would be practices (but not of a power cycle) is the concept of the Z isentropic efficiency of a prime mover. Let us assume isentropic ! (Fig. 2) that steam at pressure P1 and temperature T1 is the work produced by the turbine in the allowed to expand through a turbine to a new lower actual process 12; with final state P2; T 2 pressure P . If the expansion was reversible and adiabatic, ¼ ! 2 the work that would have been produced by the turbine i.e. isentropic, it would proceed to the lowest possible if the expansion was reversible; with final state P ; T temperature on the lower limit isobar P2, which we call T2s. 2 2 Due to the fact that the expansion through the turbine is W 1!2 h1 h2 always irreversible, i.e. some of the available work is ¼ ¼ . ð40Þ W rev; 1!2 W rev; 1!2 converted to heat, the steam, which is still expanding to the One may note that in this definition the reversible isobar P2, will end at a temperature T24T2s. In common thermodynamic engineering analysis, an expansion from 1 to 2 would take place with an entropy ‘‘isentropic efficiency’’ of the turbine is defined as increase, which can, for example, be obtained with simultaneous reversible addition of heat. Z isentropic !Although the Second Law efficiency, as defined by the work produced by the turbine in the Eq. (38) is to our opinion valuable and is in frequent use, especially in the traditional thermodynamics work in actual process 12; with final state P2; T 2 ¼ !Europe, it also has its detractors. The reservations are the work that would have been produced by the turbine because it is not easy, and sometimes impossible, to if the expansion was isentropic; with final state P2s; T 2s uniquely define the appropriate reversible system (some-

h1 h2 times called the ‘‘reversible (or ideal) model’’) that ¼ . ð39Þ corresponds to the real one that is being analyzed. Some h1 h2s examples of such model systems are discussed below. The performance of the turbine operating between state 1 (P1, T1) and state 2 (P2, T2) is compared with that which operates isentropically (‘‘ideally’’) between state 1 4.2. A starting case: a ﬂow process between states 1 and 2

Referring to Fig. 2 we examine the Second Law T efﬁciency of the process 1-2, assuming a simple case of just work output and heat input. Following Eq. (6), here the energy efﬁciency is simply P1 W u 1 ZI (41) T1 Qhp

P2 and the efﬁciency of the process 1-2 when conducted reversibly is T2 ! W T u 2s 2 Zrev ¼ , (42) Qhp rev 2s T 0 so .

W u Qhp Q . W u hp;rev ZII ¼ ¼ . (43) W u;rev Q W u;rev Qhp;rev hp

For practical reasons (and without excluding any other s uses of Eq. (43)) one most often compares the actual and Fig. 2. For expansion efﬁciency deﬁnitions. reversible processes based on the same heat input, ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 289

Qhp ¼ Qhp,rev, when Eq. (43) would then become 4.3.2. Reversible systems in which the source and sink temperature are constant W u W u ZII ¼ ¼ . (44) A reversible cycle described by Fig. 3, operating between W u;rev ðÞh2 h1 T 2ðs2 s1Þ a high temperature isotherm Th and a low temperature It is interesting now to compare this Second Law isotherm Tc, (the Carnot cycle) has a first law efficiency of efficiency with the exergy efficiency for that process. ZI ¼ Wu/Qhp , and a reversible cycle (Carnot) efficiency Using the general equation (24) for this process to Qo TcðS4 S1Þ Tc determine the exergy efficiency with a dead state choice Zrev ¼ 1 ¼ 1 ¼ 1 (51) Qi ThðS3 S2Þ Th of T0 results in (because S4 –S1 ¼ S3 –S2 by the cycle definition), and W u W u thus, from Eqs. (38), (41), and (51) u ¼ ¼ . (45) Ahp;i ðÞh2 h1 T 0ðÞs2 s1 W u ZII ¼ . (52) Obviously, the Second Law and exergy efficiencies would Qhp 1 ðT c=ThÞ be identical only if it is chosen that T0 ¼ T2. It is thus also of interest to evaluate their relative values, viz. Although it looks like the expression for the exergy efficiency of the same cycle, Eq. (30), it differs from it in u ½ðh2 h1Þ=ðs2 s1Þ T2 T ¼ , that the low isotherm c may be different, and usually is, ZII ½ðh2 h1Þ=ðs2 s1Þ To from the dead state temperature T0 chosen for the exergy analysis. 41 for T2oT o; o1 for T24To. ð46Þ

4.3.3. Example 1. For a system producing just power A power engine absorbs heat Q ¼ 2000 kJ from a high 4.3. Single output simple power cycles h temperature heat source (Th ¼ 1000 K) to produce power, and rejects Q ¼ 800 kJ to a low temperature heat sink 4.3.1. General definitions c (at T ¼ 303 K), the environment temperature T ¼ 298 K. The Second Law efficiency, Eq. (38), is used, and here c e The net power output is thus the energy efficiency is simply W u ¼ 2000 800 ¼ 1200 kJ W u ZI (47) Qhp and the energy efficiency according to Eq. (41) is and the reversible cycle efficiency is ZIu ¼ W u=Qhp ¼ 0:60. ! The plant operates between hot and cold reservoirs W u Zrev ¼ , (48) having temperature Th and Tc, respectively. The exergy Qhp rev so . T

W u Qhp Q . W u hp;rev ZII ¼ ¼ . (49) W u;rev Q W u Qhp hp Q rev i 2 3 As stated in the above section, for practical reasons one Th most often compares the actual and reversible processes based on the same heat input, Qhp ¼ Qhp,rev,, hence Eq. (43) becomes

W u ZII ¼ . (50) W u;rev Tc

To compute ZII (Eqs. (43), (50)), the cycle for which Zrev 1 4 Qo needs to be calculated must have a close relationship to the To cycle under consideration, for which ZI is known, as stated at the outset. Starting with the work by Martinovsky [19,20] and Niebergall [21], the definition for that purpose of a ‘‘reversible model cycle’’ was advanced by Morosuk s and co-workers [22,23] We now examine several cases for Fig. 3. Reversible system with constant source and sink temperatures the definition of this reversible cycle. (Carnot). ARTICLE IN PRESS 290 N. Lior, N. Zhang / Energy 32 (2007) 281–296 efficiency according to Eq. (30) is T 3 W u u ¼ ¼ 0:855. Qh 1 ðT e=ThÞ The Second Law efficiency is defined by comparison with an ideal plant working in between the same hot and cold Qi reservoirs, and according to Eq. (52) is thus:

W u ZII ¼ ¼ 0:861. 4 Qhp 1 ðTc=ThÞ Because of the definition-based use of different reference 2 systems, the exergy efficiency and the Second Law Q efficiency are different. They are identical to each other o only for the specific case when the heat sink temperature Tc 1 T0 is identical to the environment temperature Te.

4.3.4. Reversible systems with varying source and sink temperatures s In many power cycles the heat input and rejection is not Fig. 4. Reversible systems with varying source and sink temperatures even approximately isothermal. The reversible power cycle (Lorenz). named after Lorenz [24] (Fig. 4) can serve as the ‘‘reversible (or ideal) model cycle’’ in such cases, and also serves well to introduce the ‘‘entropic temperature’’ concept useful in between states 1 and 4, as well as between 2 and 3, are such analyses. This cycle consists of an isentropic negligible, is compression stroke 1-2, followed by a stroke 2-3in Q4!1 m_ cðÞh4 h1 which heat is transferred to the working fluid at gradually Zrev ¼ 1 ¼ 1 , (56) Q2!3 m_ hðÞh3 h2 increasing temperature till the top temperature T3 of the cycle is reached, followed by an isentropic expansion stroke where for generality it is assumed that the mass flow rates 3-4 to a lower pressure, and closed with a heat rejection in the heated and cooled strokes are different, m_ h and m_ c, stroke 4-1 at variable temperature. One case of the respectively. It is important to note at this point that the Lorenz cycle is the Brayton cycle (called by some the Joule heat input Q2-3 and the heat rejection Q4-1 may either be cycle) in which the heat input and rejection strokes take calculated as the enthalpy changes of the working fluid of place isobarically. the cycle itself (then usually, when the working mass flow The equation of the first law efficiency of the Lorenz rate is the same in strokes 2-3and4-1, with, say, value cycle remains the same as before, (41), but the reversible m_ ; m_ c ¼ m_ h ¼ m_ ), or as the enthalpy changes of the heat cycle efficiency, Zrev , defined as source and sink fluids. In the latter case, the corresponding R temperatures of the heat source fluids must be higher than W Q Q Q 4 T dS u i o o R1 the cycle fluid, Tf,2-34T2-3, and of the heat sink fluids Zrev ¼ ¼ ¼ 1 ¼ 1 3 (53) Q Q Q must be lower than the cycle fluid, Tf,4-1oT4-1, to allow i i i 2 T dS heat transfer. It is also noteworthy that Qf,2-3 ¼ Q2-3 and is a little more complex to calculate. A conceptual Qf,4-1 ¼ Q4-1 only if there are no heat losses in these heat simplification is to express the efficiency of the Lorenz transfer processes. cycle in terms of an equivalent Carnot cycle operating Because S4 –S1 ¼ S3 –S2 by the cycle definition, between the temperatures T^ h and T^ c, also known as the m_ s s entropic temperatures, defined by c ¼ 3 2 . (57) R R 3 4 m_ h s4 s1 2 T dS 1 T dS T^ h T^ c . (54) so substituting Eq. (57) into Eq. (56) gives S3 S2 S4 S1 ðÞh h =ðÞs s It is easy to show that the Carnot cycle operating between Z ¼ 1 4 1 4 1 (58) rev ðÞh h =ðÞs s the temperatures T^ h and T^ c has the same efficiency as the 3 2 3 2 Lorenz cycle described in Fig. 4, and the Lorenz cycle and thus if the average temperatures of the Lorenz cycle efficiency is thus are defined by Eq. (58) in the form of Eq. (54), we get ^ T c h3 h2 h4 h1 Zrev;Lorenz ¼ 1 . (55) T^ h ; T^ c (59) T^ h s3 s2 s4 s1 An alternate way to express Eq. (53), when the changes in and the efficiency of the Lorenz cycle is again expressed by the kinetic and potential energy, and work interactions, Eq.(55). ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 291

Further simpliﬁcation of Eq. (59) is possible, by Using Eqs. (43), (53), and (55), the Second Law efﬁciency expressing the enthalpies and entropies by the directly based on the Lorenz cycle can be expressed as measurable properties such as T and p: W u @v ZII ¼ dh ¼ C dT þ½v Tð Þ dp (60) Qhp 1 Q4!1=Q2!3 p @T p W ¼ hiR u . R 4 3 and Qhp 1 1 T dS 2 T dS W dT @v ¼ u ð69Þ ds ¼ CP ð Þ dp. (61) ^ ^ T @T P Qhp 1 T c=Th

Integration of Eqs. (60) and (61), and substitution into ZII defined by Eq. (69) is equal to the exergy efficiency e of Eq. (59) gives that system only if it is chosen that T0 ¼ T^ c. R R T3 p3 @v ^ Cp dT þ ½v Tð Þ dp 4.4. Single output refrigeration cycles T2 p2 @T p T h ¼ R R , (62) T3 Cp dT p3 ð @v Þ dp T2 T p2 @T p 4.4.1. COP-based Second Law efficiency Refrigeration cycles are traditionally defined by the R R coefficient of performance [COP, Eq. (8)]. Consider T4 p4 @v ^ Cp dT þ ½v Tð Þ dp a general refrigeration system operating on the paid T1 p1 @T p T c ¼ R R . (63) T4 Cp p4 @v energy inputs of heat Qhp at temperature Ti and work dT ð Þp dp T1 T p1 @T Wp, to produce cooling of magnitude Qcu, and heat Qo is rejected at temperature To, as described in Fig. 5. The

In the case when Cp ¼ const, Second Law efﬁciency based on the COP would thus be R deﬁned as p3 @v ^ CpðT3 T 2Þþ p ½v Tð@T Þp dp . T ¼ 2 R , (64) h p3 @v Cp lnðT 3=T 2Þ ð Þ dp Qcu W p þ Qhp p2 @T p COP hi. COPII ¼ COPrev Qcu W p þ Qhp R rev p4 @v ^ C ðT T Þþ ½v Tð Þ dp p 4 1 p @T p W p þ Qhp T ¼ 1 R . (65) Qcu rev c p4 @v ¼ 1. ð70Þ Cp lnðT4=T1Þ p ð@T Þp dp Q 1 cu rev W p þ Qhp

A further simpliﬁcation is gained if the ﬂuid is an ideal gas, The comparison between the actual and reversible cycles is following pv ¼ RT and thus ð@v=@TÞp ¼ R=p ¼ v=T, then Eqs. (64) and (65) become often made based on the same useful cooling output for C ðÞT T T^ ¼ p 3 2 , h Q Cp lnðT 3=T 2ÞR lnðp3=p2Þ hp CpðÞT4 T 1 T^ c ¼ . ð66Þ Cp lnðT 4=T 1ÞR lnðp4=p1Þ

Further, if the heat absorption and ejection processes are under constant pressure (like in the Brayton cycle), then dp ¼ 0. if Cp is constant, then we can get the following Absorption equation: Q Wp 0 cooler

T 3 T 2 T4 T 1 T^ h ¼ ; T^ c ¼ . (67) lnðT3=T2Þ lnðT4=T1Þ

In this particular case, the efﬁciency of the Lorenz cycle is expressed as Q ðÞT4 T 1 = lnT 4=T 1 cu Zrev ¼ 1 . (68) ðÞT3 T 2 = ln T 3=T 2 Fig. 5. Energy inputs and outputs for an absorption cooler. ARTICLE IN PRESS 292 N. Lior, N. Zhang / Energy 32 (2007) 281–296 both, Qcu ¼ (Qcu)rev, and then Eq. (70) becomes reversible cycle as Q 1 W p þ Qhp ¼ cu rev Qhp W p . (79) COPII ¼ 1. (71) COP~ c Z~hp W p þ Qhp

Substituting Eq. (79) in the denominator of the COPrev After choosing an appropriate ideal model cycle one can (shown in Eq. (70)) gives calculate COPrev (or the numerator of Eq. (71)) and thus 2 3 COPII when using Eqs. (70) or (71), respectively. A direct 4Qcu 5 way leading to equations that allow the computation of COPrev ¼ (80) ~ Qcu=COPc Z~hp W pð1=Z~hp 1Þ COPrev (and thus COPII) without having to choose ideal rev model cycles, is based on the application energy and entropy balances to the system of Fig. 5, and several and based on Eqs. (70) and (80) the Second Law COP is assumptions as shown below. From energy conservation hi Q =COP~ Z~ W ð1=Z~ 1Þ Q þ Q þ W p Q ¼ 0. (72) Q cu c hp p hp hp cu o COP ¼ cu rev II Q cu rev Qhp þ W p The entropy balance for the reversible cycle DS ¼ 0, thus we can write (81) Qhp Q Q simplifying, if we assume for comparison that Q ¼ Q þ cu o ¼ 0. (73) cu cu,r- T i Tc T o rev ev,to ~ We note at this point that if these heat source and sink Qcu=COPc Z~hp W p;revð1=Z~hp 1Þ temperatures are not constant, the entropic temperature COPII ¼ (82) Q þ W definition, Eq. (54), (or Eqs, (59) and (67)) can be used. hp p Rearranging, All the terms on the right-hand side of this equation, except Qo ¼ W p þ Qhp þ Qcu. (74) the required reversible work input Wp,rev, are known. If the latter can be estimated, COPII can be calculated. One way To simplify, and without loss of generality, it is reasonable to eliminate Wp,rev from the equation is to express it in to assume that the heat rejection is to the environment at terms of Qhp by assuming, for example, that the real and the temperature of the environment, Te,soTo ¼ Te and reversible cycle have the same Wp/Qhp ratio. using this and Eq. (74) in Eq. (73) gives Applying Eqs. (80) and (81) to the specific cases where the refrigeration cycle has only a work input (such as in Q Q Q þ Q þ W p hp þ cu hp cu ¼ 0 (75) vapor compression systems) Ti T c Te T c COPrev ¼ COP~ c ¼ (83) T e Tc or and T T T Q e c e ¼ COP Te Tc cu Qhp 1 W p 0. (76) COPII ¼ ¼ COP . (84) T c T i COP~ c Tc In another case when the required work input is negligible The terms in the parentheses are now named the reversible relative to the heat input (such as in absorption refrigera- cycle efficiencies as follows: tion systems),

Te ~ T c T i T e Z~hp 1 , (77) COPrev ¼ COPc Z~hp ¼ . (85) Ti T e Tc T i That is a well-known expression, which for the more general case where the heat rejection temperatures in the T c COP~ . (78) absorber and condenser of the absorption cycle, Ta and c T T e c Tcon, are not identical, but under the idealized operation assumption that the solution concentration remains con- Using Eqs. (76)–(78) we can express the paid heat and work stant (inﬁnitesimally small evaporation rates) and ideal inputs required to produce the useful cooling output of the heat recovery between the generator and absorber, making ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 293

Qa ¼ Qhp, becomes (cf. [21]) 4.5. Systems that simultaneously produce useful work, cooling and heating ð1=T aÞð1=T iÞ COPrev ¼ (86) ð1=T cÞð1=TconÞ In that case, and abbreviating the notations, and corresponding to Eqs. (85) and (86) the equations in W u þ Qhu þ Qcu ZI ¼ ¼ Zw þ ZQ þ COPc, (90) this case for COPII are Qhp COP COP ¼ (87) where Wu,Qhu, and Qcu are the useful outputs of work, II ~ COPc Z~hp heat and cooling, Qhp is the paid heat input to the cycle, and the efﬁciency terms on the right-hand side of the and equation are deﬁned as, the useful work production

ð1=T cÞð1=TconÞ efﬁciency, COPII ¼ COP , (88) ð1=T aÞð1=T iÞ W u Zw , (91) respectively. Qhp They differ from the expressions for the exergy COP of the useful heat production efﬁciency, the same cycle, Eq. (31), in that the temperatures may be Q different, and usually are, from the dead state temperature Z hu (92) Q Q T0 chosen for the exergy analysis. hp and the useful cooling output COP,

4.4.2. Example 2. For a system producing just cooling Qcu In a vapor compression cooling system, the working COPc . (93) Qhp fluid absorbs heat Qcu ¼ 600 kJ from a low temperature It is noteworthy that all these efficiencies are normalized by heat source (at Tc ¼ 200 K), and rejects heat Qe ¼ 1000 kJ the same total paid heat input Q ignoring the fact that to the environment, Te ¼ 298 K, hp The net power consumed is thus W p ¼ 1000600 ¼ only some fraction of that heat input produces each of 400 kJ, and so from Eq. (9): these outputs. It is often of practical utility to define the ratios of the COP ¼ Qcu=W p ¼ 1:5. cooling and heating energy outputs to the work output, as

If we assume that the dead state temperature T0 is the Qcu same as the temperature of the environment, T , the rc , (94) e W u exergy efﬁciency according to Eq. (31) is u ¼ QcuðT e T cÞ=T c =W p ¼ 0:735:According to Eqs. (83) Qhu rh (95) and (84), COPrev ¼ Qcu=W p;rev ¼ Tc=ðTe TcÞ¼2.04, and W u

COPII ¼ COP=COPrev ¼ 0:735. and then Eq. (90) takes the form ¼ ðÞþ þ It is found that the COPII is the same as the exergy ZI 1 rc rh Zw. (96) efficiency, because the system dead state temperature is Eq. (96) is useful when comparing efficiencies of chosen to be the environment temperature Te; they would systems that have the same values of the output heat-work be different if the choice was different. ratios r. The reversible system efficiency can be expressed similarly as 4.4.3. Reversible systems with varying source and sink ! temperatures W u þ Qhu þ Qcu As in the case of power cycles discussed in Section 4.3.3, Zrev ¼ ¼ Zw;rev þ ZQ;rev þ COPc;rev, Qhp in many cooling cycles the heat input and rejection is not rev even approximately isothermal. The cooling cycle named (97) after Lorenz (the reverse of the cycle in Fig. 4) is similarly where the efficiency terms on the right-hand side of the used, and the entropic temperatures, defined for example equation, all for the reversible case, are defined as follows: by the useful work production efficiency, R R 3 T dS 4 T dS ^ 2 ^ 1 W u;rev T h ; T c (89) Zw;rev , (98) S3 S2 S4 S1 Qhp;rev are substituted for the temperatures used in the equations the useful heat production efficiency, in Section 4.4.1. Qhu;rev Simplifications of the entropic temperatures can be used, ZQ;rev (99) if appropriate, as those similar to Eqs. (54), (59) and (67). Qhp;rev ARTICLE IN PRESS 294 N. Lior, N. Zhang / Energy 32 (2007) 281–296 and the useful cooling output COP, Qcu r~c , (110) W u Qcu;rev rev COPc;rev . (100) Qhp;rev Qhu r~h , (111) Similar to the energy balance shown in Fig. 1 but W u rev somewhat simplified for clarity, let us assume that a system Q produces useful work amount Wu, useful cooling Qcu at i;e r~e (112) temperature Tc, and useful heat Qhu at temperature To,h, W u rev has a paid heat input Qhp at temperature Thp, and has and with the substitution of Eqs. (110)–(112), Eq. (109) ‘‘free’’ heat output to- and input from- the environment, takes the form Qo,e and Qi,e at temperatures Ti,e and To,e, respectively. As done in Section 4.4.1, the entropy balance for this Z~hpðÞ1 þ r~c þ r~h rev Zrev ¼ . (113) reversible cycle is 1 þðr~c=COP~ cÞþZ~ r~h þðr~e=COP~ eÞ hu Q Q Q Q Q The Second Law efficiency is thus, using Eqs. (38), (90) and hp þ cu þ i;e hu o;e ¼ 0 (101) (97) T hp Tc T i;e T hu To;e rev and energy conservation Zw þ ZQ þ COPc ZII ¼ , (114) Zw;rev þ ZQ;rev þ COPc;rev Qhp þ Qcu þ Qi;e Qo;e Qhu W u ¼ 0. (102) or from Eqs. (96) and (113) gives another equation for ZII We note at this point that if these heat source and sink which is often easier to use: temperatures are not constant, the entropic temperature definition, Eq. (54) can be used. Combining Eqs. (101) and ZwðÞ1 þ rc þ rh r~c r~e ZII ¼ 1 þ þ Z~hur~h þ . (102) to eliminate Qo,e yields Z~hpðÞ1 þ r~c þ r~h COP~ c COP~ e T T T T (115) Q 1 o;e þ Q o;e c þ Q 1 o;e hp T cu T hu T hp c hu All of the terms in Eq. (115) except r~e are known from T o;e T i;e either choice or calculation, so if r~e can be determined by þ Qi;e þ W u ¼ 0. ð103Þ some other means, Z can be calculated. T i;e II Further simplification of efficiency comparisons is The terms in the parentheses are now named the reversible obtained if it is chosen that the heat-power outputs in the cycle efficiencies as follows: actual and reversible cycle are the same, rc þ rh ¼ r~c þ r~h, T and then Eq. (115) becomes Z~ 1 o;e , (104) hp T hp Zw r~c r~e ZII ¼ 1 þ þ Z~hur~h þ . (116) Z~hp COP~ c COP~ e T c COP~ c , (105) T o;e Tc If it can be assumed that the heat exchange with the environment occurs at the temperature of the environment, To;e Te,soTi,e ¼ To,e, Eq. (116) becomes Z~hu 1 , (106) T hu Z r~ Z ¼ w 1 þ c þ Z~ r~ (117) II Z~ ~ hu h T i;e hp COPc COP~ e . (107) T o;e Ti;e and now ZII can be calculated without knowing r~e. Using Eqs. (103)–(107) we can express the paid heat It is of interest to compare these Second Law efficiency input required to produce the useful work, cooling and equations with those defining the exergy efficiency pre- heating outputs of the reversible cycle as sented in Section 3. Qcu Qi;e 1 4.5.1. Example 3 Qhp ¼ W u þ þ Z~huQhu þ . (108) COP~ c COP~ e Z~hp This example demonstrates one possible error that can be made when the exergy and Second Law efficiencies are Replacing the denominator in Eq. (97) with the last used interchangeably, without realizing their distinct expression gives definitions, using the case of a compound cycle for W u þ Q þ Q cogeneration of power, heating and cooling proposed by Z ¼ hu cu rev . rev ~ ~ the authors [25]. The plant operates in a parallel combined W u þðQcu=COPcÞþZ~huQhu þðQi;e=COPeÞ ð1=Z~hpÞ cycle mode with an ammonia–water Rankine cycle and an (109) ammonia refrigeration cycle, interconnected by the absorp- Similar to the derivation of Eq. (96), we define for the tion, separation and heat transfer processes. It is driven by reversible cycle the heat-work output ratios as one external heat source fluid. In this paper, the heat source ARTICLE IN PRESS N. Lior, N. Zhang / Energy 32 (2007) 281–296 295

fluid is chosen to be air, entering the system at 465 1C/ and the Second Law efficiency of the system is 1.043 bar, and the net power output Wu ¼ 719 kW. [Eq. (38)] Both refrigeration and heating output are at variable ZII ¼ ZI=ZI;rev ¼ 58:7%. temperatures. The refrigeration output Qcu ¼ 266.2 kW, the working fluid (rich ammonia stream) provide refrigera- In this case, the Second Law efficiency is the same as tion in a evaporator at the temperature range of 22.7 1C the exergy efficiency. to 15 1C and pressure of 1.6 bar, with the corres- (2) If the heat sink temperature is chosen to be at a ponding entropic temperature (Eq. (59)] Tc ¼ 250.5 K. value different than the environment temperature Te, The heating fluid provide low level heat at the temperature for example, if the heat sink is sensibly assumed to range of 90–50 1C and pressure of 1.013 bar, the heat be the cooling water at T0 ¼ 303.15 K, then Qhp, output Qhu ¼ 308.9 kW, at the corresponding entropic rev ¼ 2108.4 kW, and the first law efficiency of the temperature Thu ¼ 342.8 K. The heating fluid finally reversible system is thus: exhausts to the environment (T ¼ 298.15 K), and the heat e W u þ Qhu þ Qcu addition process is also a temperature variable process, ZI;rev ¼ ¼ 61:4% Qhp;rev at the entropic temperature Thp ¼ 492.5 K. The total heat and then the Second Law efficiency of the system is input Qhp ¼ 3,496 kW. The first law efficiency [from Eq. (6)]: ZII ¼ ZI=ZI;rev ¼ 60:3%; 2:7% higher than the exergy efficiency despite the smallness of the difference between W u þ Qhu þ Qcu ZIu ¼ ¼ 37:0%. (118) the temperatures of the reference states (5 K, 1.6%). Qhp

Assuming that the exergy dead state temperature T0 ¼ Te, 5. Conclusions and recommendations the refrigeration exergy output is [Eq. (29)] There are many performance criteria for energy systems T T A ¼ Q e c ¼ 50:6kW: (119) and they need to be defined and used with great care. cu cu T c The situation becomes especially complex when simul- The heating exergy output is [Eq. (28)] taneous energy interactions of different types, such as work, heating and cooling, take place with a system. T e Ahu ¼ Q 1 ¼ 40:2 kW. (120) The distinction between exergy and Second Law hu T hu efficiencies is not clearly recognized by many. The exergy input is It is attempted here to clarify the definitions and use of energy and exergy based performance criteria, and of T e Ahp ¼ Q 1 ¼ 1379:6 kW, (121) the Second Law efficiency, with an aim at the advance- hp T hp ment of international standardization of these important so, the exergy efficiency is [from Eq. (24)]: concepts.

W u þ Ahu þ Au Equations defining first law efficiencies including those u ¼ ¼ 58:7%. (122) considering embodied energy, environmental impact, Ahp and economics, and defining exergy and Second Law To calculate the Second Law efficiency, we should first find efficiencies for systems producing power, cooling, and the first law efficiency of a reversible cycle, which operates cogeneration of power, heating and cooling are pre- under the same thermophysical conditions and, consistent sented. with the derivations in this paper, we further choose one Examples demonstrate some of the magnitude differ- that has the same power, heating and cooling outputs. ences of exergy and Second Law efficiencies and the Using Eqs. (101) and (102) for this case: errors that can be made if the equations and systems are Q Q Q Q not defined carefully. hp;rev þ cu hu e;rev ¼ 0, T hp Tcu Thu T e þ ¼ þ þ ð Þ Qhp;rev Qcu Qhu W u Qe;rev, 123 Acknowledgement where Qe,rev is the heat amount exhausted reversible to the chosen heat sink. The second author gratefully acknowledges the support of the Chinese Natural Science Foundation Project (1) If the heat sink is chosen to be the environment at (No. 50576096). T0 ¼ Te ¼ 298.15 K, then we can calculate from Eqs. (123) that Qhp, rev ¼ 2052.1 kW, and the first law Appendix A. A simple example of important differences in efficiency of the reversible system is thus: energy efficiency definitions W þ Q þ Q Z ¼ u hu cu ¼ 63:1% Take two ways that the energy efficiency is define, one, I;rev Q hp;rev Z1u [Eq. (6)] subtracts the invested work (such as pumping, ARTICLE IN PRESS 296 N. Lior, N. Zhang / Energy 32 (2007) 281–296 etc.) in the numerator, and the other, Z1u,d, adds it to the [9] Grassman P. Zur allgemeinen Definition des Wirkungsgrades (to the denominator instead. In the simplest form, the definitions general definition of the efficiency). Chem Ing Technol 1950; are 22(4):77–80. [10] Nesselman K. U¨ber den thermodynamischen Begriff der Arbeitsfa¨- W W higkeit (on the thermodynamic concept of work potential). Allg Z ¼ u p , (124) 1u Q Wa¨rmetechn 1952;3(5/6):97–104. hp [11] Nesselman K. Der Wirkungsgrad thermodynamischer Prozesse (the efficiency of thermodynamic processes). Allg Wa¨rmetechn 1953;4(7): W u 141–7. Z1u;d ¼ , (125) Qhp þ W p [12] Fratzscher W. Zum Begriff der Exergetischen Wirkugsgrades (to the concept of the exergetic efficiency). Brenns Wa¨rme-Kraft 1961; It is of interest to know which of the definitions gives 13(11):486–93. larger efficiency values than the other, so we examine the [13] Rant Z. Bilanzen und Beurteilungsquotienten bei technicshe prozes- ratio sen (balances and evaluation coefficients of technical processes). Int Z ! Gaswa¨rme 1965;14(1):28–37. [14] Baehr HD. Zur Defintion exergeticshe Wirkuksgrader (to the ZIu ðW u W pÞ=Qhp W p W p ¼ ¼ 1 1 þ definition of exergetic efficencies). Brennst Wa¨rme-Kraft 1968;20(5): Z W u=ðQ þ W pÞ W u Q Iu;d hp ! hp 197–200. W 2 [15] Dunbar WR, Lior N, Gaggioli R. The component equations of 1 1 p energy and exergy. ASME J Energy Resources Technol 1992;114: ¼ 1 þ W p . ð126Þ Qhp W u W uQhp 75–83. [16] Dunbar WR, Lior N. Sources of combustion irreversibility. Combust The last term is thus always negative, and the term Sci Technol 1994;103:41–61. [17] Lior N. Irreversibility in combustion, invited keynote paper. before last is negative when Qhp4Wu. Since the latter is always true thermodynamically, the conclusion is that Proceedings of ECOS ‘01: efficiency, costs, optimization, simulation and environmental aspects of energy systems, vol. 1, Istanbul, Z1u/Z1u,do1. Thus one can say that defining the efficiency Turkey, 2001. p. 39-48. as Z1u is a more conservative evaluation of the efficiency, [18] Lior N, Sarmiento-Darkin W, Al-Sharqawi HS. The exergy fields in favoring in financial negotiations the system customer, and transport processes: their calculation and use. invited keynote presentation and paper, Proceedings of the ASME — ZSIS Z1,d gives a higher value, thus favoring the system vendor. international thermal science seminar, vol. II, Bled, Slovenia, June 13–16, 2004. pp. 155-69; expanded version published in Energy 2006; References 31:553–78. [19] Martinovsky VS. Thermodynamic characteristics of heat and [1] Lior N. Research and new concepts. In: Lo¨f GOG, editor. Active refrigeration machines cycles. Moscow-Leningrad: Government solar systems. Cambridge MA: MIT Press; 1993. p. 615–74 [Chapter Energy Publication; 1952. 17]. [20] Martinovsky VS. Cycles, schemes and characteristics of thermo- [2] Vijayaraghavan S, Goswami DY. On evaluating efficiency of a transformers. Moscow: Energy; 1979. combined power and cooling cycle. Trans ASME J Energy Resources [21] Niebergall W. Sorptions-Ka¨ltemaschinen. In: Plank R, editor. Technol 2003;125(3):221–7. Handbuch der Ka¨ltetechnik, vol 7. Berlin: Springer; 1959. [3] White SW. ‘‘Birth to death’’ analysis of the energy payback ratio and p. 23–30.

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