LOST WORK: A MEASURE OF THERMODYNAMIC EFFICIENCY

NOEL DE NEVERS and J. D. SEADER  Department of Chemical Engineering, University of Utah, Salt Lake City, UT 842 12, U.S.A. Abstract- Th...

NOEL DE NEVERS and J. D. SEADER 
Department of Chemical Engineering, University of Utah, Salt Lake City, UT 842 12, U.S.A.
Abstract-The introduction of “lost work” into the statement of the second law of thermodynamics allows us to combine it with the lirst law to obtain a statement of extreme breadth and generality. Because the combined tirst- and second-law statement defines reversible work, all other widely used statements of minimum or maximum work can be shown to be restricted cases of this combined statement. With this combined statement, we can calculate thermodynamic efficiencies of all processes, including not only those conventionally treated-which are large work producers or consumers-but also those that do not have work production or consumption as their goals (as in absorption refrigerators or distillation columns). The results obtained this way, for example with turbines, are not the same as the conventional “isentropic efficiency” definition, but are more thermodynamically sound and much more practical for turbines whose outlet temperatures are far removed from the ambient temperature. The combined first- and second-law statement leads naturally to the availability function and the batch availability function rather than to the availability. For practical process problems, where prime movers are not the most important concern, the availability function is a much more useful and practical quantity to utilize than the availability.

INTRODUCTION The second law of thermodynamics is not new. It has been known in all of its important aspects since at least 1870. The objective of this paper and this conference is to seek out the most satisfactory ways of applying the second law to practical problems. Although the public is slowly being educated to think about “the energy crisis”, we technical people understand perfectly well that the real problem is “the entropy crisis”. The energy of the earth is changing very little, if at all, and if solar radiation and outgoing heat flux are balanced, it is not changing at all. Energy is conserved (i.e. the first law is obeyed) in all of our most wasteful uses of fuels and electricity (e.g. refrigerated swimming pools in hot climates, the lights of Las Vegas, etc.). Our problem is that our low-cost sources of low-entropy materials from which we can extract useful work to drive our vehicles or to power our factories or to heat our homes have become inadequate for our current consumption. We will probably be well advised not to confuse the public by telling them about the “entropy crisis”, but certainly all technical people must be aware of this. The purpose of this conference is to evaluate various ways of applying the second law of thermodynamics to the calculation of thermodynamic efficiencies, There have been many proposals, of which the most commonly used method is “availability analysis”.‘-3 In this paper, we extend the development of a less well-known approach involving “lost work”. If all of these methods are used properly, they must all give the same answer since the second law is unambiguous. Thus, the criteria for selecting the best procedure to evaluate thermodynamic efficiency should be: (1) best ease of use, (2) best degree of correspondence with the viewpoint and background of intended users, and (3) greatest breadth of application. On these grounds, we believe that the lost work approach is superior to other approaches in common use. 

THE SECOND LAW OF THERMODYNAMICS AS AN ENTROPY BALANCE WITH LOST WORK 
Classical formulations of the second law of thermodynamics take the form of inequalities. The first form proposed was for closed systems where Qi refers to the heat transfer across a portion i of the boundary of the system and Ti is the corresponding temperature of that part of the boundary. Except for isothermal systems, T is not uniform, but has different values at different parts of the boundary. As will be shown later, the selection of the system, and therefore its boundary, is important and requires careful consideration. In general, so that all irreversibilities occur within the system, the boundary is placed such that values of Ti correspond to the various heat reservoir temperatures. For open systems, we must include the possibility of material flowing across portions j of the system boundary. Thus In both equations, the summations are algebraic so that a flow out of the system has a negative value of Q or m. Here, and in subsequent equations, the A(ms& implies either that the system has uniform intensive properties or that the product of mass times the specific entropy (or other intensive property functions used later) is obtained by an integration over the entire mass contained within the system. In addition, the C(ms)j and X(QJTi) terms imply either that the flows across the boundaries are uniform with respect to all variables or that the terms correspond to integrations over the boundaries of the system. The details of such integrations are given by Bird.4 A more useful equality form of Eq. (2) is obtained by adding a term for the irreversible entropy increase, to give Denbigh’ stated that introducing such an irreversible entropy increase term to convert the second law from an inequality to an equality is largely due to de Donder. Although Denbigh also utilized an equality form of the second law, he applied it to both the system and the surroundings. As shown later, the procedure adopted here leads to an equation that is more readily applied than the equation developed by Denbigh. Any discussion of thermodynamic or second-law efficiency must be concerned with ASi,,. If it is zero, the process is reversible and is as efficient as is theoretically possible. The larger ASi,, all other things being equal, the less efficient the process. This form of the second law also allows US to formulate this neat verbal statement of it: ASi, is positive in the real world, zero in the theoretical reversible world, and negative in some impossible worlds. The main idea of this paper is to show that, if we define a new quantity called “the lost work” as and introduce it into Eq. (3), we will have a very powerful and convenient method of assessing the thermodynamic efficiency of processes. In this equation, T,, is the temperature of the infinite surroundings, normally taken as the temperature of the nearest large body of ambient water like a river, lake, or ocean, Because they are assumed to be infinite in size, infinite surroundings are an ultimate heat source or sink that is always available and does not need to be restored or replenished. Although one could work totally with ASi,, or perhaps mentally substitute T&Sirr whenever LW appears, we believe that formulations in terms of LW are simpler, easier to use, and more intuitively satisfying than any other formulation of the second law now in common usage. Indeed, as we show later, lost work is exactly what the term implies-work that is irreversibly lost. Table 1 lists the most common ways in which irreversible entropy increases (ASi,) occur. Conceptually, all of these are equivalent to the conversion of work to heat. Hence, at least as early as 1950: the idea of “lost work” had been used as a measure of irreversibility or of the degradation of energy from more useful to less useful forms. As we discussed in an earlier paper,’ there are two definitions of lost work in common usage. The first one indicates that lost work is simply the amount of work that is irreversibly converted to heat. The other, which we consider the superior definition, indicates that lost work is not only the amount of work that is irreversibly converted to heat, but also includes the additional work that must be transferred to the system to offset the consequences of this degradation. This idea is illustrated in Fig. 1. Here, we see an irreversible process occurring in a system that may be open or closed, where the irreversible work input is lost. In addition, a Carnot refrigerator offsets this irreversible work conversion by removing the incremental heat and rejecting it to the surroundings at To so as to restore the system to its original state. Lost work, as we believe it should be defined, is the sum of these two works. One may also show that lost work is equivalent to the incremental heat that must be rejected to the infinite surroundings at T,, due to irreversibilities (see Appendix 1). Since thermal energy in the heat reservoir at T,, cannot be put to any practical use, this is energy that has been degraded to its ultimate uselessness. 

THE ADVANTAGES OF THE LOST WORK CONCEPT 
(1) The lost work concept allows us to formulate the second law in an equality form rather than in an inequality form 
AS =A(ms),,,=~~+~(,,).+~. sys iZ i ’ 0 
(5) 
This is merely a restatement of Eq. (3), in which the ASi, term is replaced by Eq. (4). However, this substitution replaces a term of low intuitive content with two others. One is an external variable that changes little from location to location, and the other is directly related to external variables and is the term of practical significance. Lost work is the equivalent of the incremental extra electric power that must be purchased to offset irreversibilities, but there is no such simple intuitive interpretation of ASir,. (2) We can multiply Eq. (5) by TO, subtract it from the general energy balance, and then rearrange to produce the combined statement form 
(‘W,“W)=X[m(h-To~)lj+~ (I-~)Q-b[m(~-~~~)l,,,. (6) I 
Here we have followed the thermodynamic sign convention that heat flowing into the system is positive, and work flowing out of the system is positive. We can further simplify this result by introducing a combination of thermodynamic variables, often defined on a unit mass or mole basis as the “availability function”, or 
b=h-Tos=u+Pv-Tos. (7) 
This function and symbol-but not name-were apparently introduced into engineering by Keenanr’ although it had been proposed earlier by others in different forms? Haywood calls it the “steady-flow availability” function, while Faires and Simmang” refer to it as the “availability” or “Darrieus” function. However, since “availability function” seems to be its most common name, it will be used here. It is unfortunate that the word “availability” appears in its name, however, since it leads to confusion with what many authors,**“*” in writing of a steady-state, steady-flow process, have called the “availability” or (h - Tos) - (ho - Toso) in the absence of potential and kinetic energy effects Substituting Eq. (7) into Eq. (6), we obtain 
(CW,+LW)=z(mb)j+T (1-~)8_P[m(b_Pv)l,v,. J I 
The term on the far right contains the combination (b - Pv), which is logically called a “batch” or “nonflow” availability function. Haywood’ suggests that it should have the symbol a, but, to our knowledge, no symbol or name has been widely adopted for it. We can easily divide Eq. (8) by the time change At to obtain the equivalent form in terms of power, mass flow rates, heat flow rates, and time rate of change of the system. (3) We can see that, in this combined statement, the two work terms of actual external work and lost work appear together. Their sum is equal to the reversible work of a process that has the same material flows in and out and the same heat exchange at all temperatures other than T,, so that 
W,,,=ZW,+LW. (9) 
This is an extremely broad and powerful result. It means that, for any real process where we can define the state of the system before and after any system change, the mass flows or mass flow rates into and out of the system, and the heat flow rates and temperatures at which heat flows into and out of the system, we can directly and unambiguously compute the work flows for a reversible process. From the difference between that work flow and the actual work flow, we can compute the lost work. (4) With this lost work definition, we can easily compute thermodynamic efficiencies for processes that involve exchange of work with the surroundings and for those that do not. For processes that exchange work with the surroundings, and whose purpose is either to produce work or to consume work, the definitions are simply 
(10) 
(11) 
(5) We can see that efficiency definitions given in Eqs. (10) and (11) are not the same as the ones commonly shown in many engineering thermodynamics textbooks. For example, Eq. (10) does not lead to the conventional definition of the “isentropic efficiency” of a turbine, but to a much more powerful definition. The conventional definition, according to Van Wylen and Sonntag,” is 
W h,-h2 =e=- qturbine mechanical w, h, _ h2,) (12) 
which is a mechanical efficiency definition. The initial state and the final pressure of the real and reversible (isentropic) processes are chosen to be the same. This fixes the outlet temperature and entropy of the isentropic process. The outlet temperature of the real turbine process can be arbitrarily estimated or determined experimentally. We may see this from Fig. 2, where a conventional Mollier diagram representation of the process path of an adiabatic steam turbine with negligible kinetic energy changes is shown. The initial state is 1; the final actual state is 2; and the reversible path leads from 1 to 2,. In contrast, the definition which follows from Eq. (9) is a thermodynamic definition, where the inlet and outlet states are the same for both the real and the reversible turbines. If we apply Eq. (10) to the same system, we see that 
%urbine thermodynamic 
=e h-h_ =-_ hl-hz W e,, h-h h,--hz- Z-ob,- ~2)’ 
(13) 
To see the relation between the thermodynamic turbine efficiency and the more common mechanical turbine efficiency, we define 
Tavg = h2 - h2, s2-82,’ 





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