The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 3

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Expressed analytically, the Kelvin-Planck statement isWcycle ≤ 0(single reservoir)where the words single reservoir emphasize that the system communicates thermally only with a singlereservoir as it executes the cycle. The “less than” sign applies when internal irreversibilities are presentas the system of interest undergoes a cycle and the “equal to” sign applies only when no irreversibilitiesare present.IrreversibilitiesA process is said to be reversible if it is possible for its effects to be eradicated in the sense that thereis some way by which both the system and its surroundings can be exactly restored to their respectiveinitial states.

A process is irreversible if there is no way to undo it. That is, there is no means by whichthe system and its surroundings can be exactly restored to their respective initial states. A system thathas undergone an irreversible process is not necessarily precluded from being restored to its initial state.However, were the system restored to its initial state, it would not also be possible to return thesurroundings to their initial state.There are many effects whose presence during a process renders it irreversible. These include, butare not limited to, the following: heat transfer through a finite temperature difference; unrestrainedexpansion of a gas or liquid to a lower pressure; spontaneous chemical reaction; mixing of matter atdifferent compositions or states; friction (sliding friction as well as friction in the flow of fluids); electriccurrent flow through a resistance; magnetization or polarization with hysteresis; and inelastic deformation.

The term irreversibility is used to identify effects such as these.Irreversibilities can be divided into two classes, internal and external. Internal irreversibilities arethose that occur within the system, while external irreversibilities are those that occur within thesurroundings, normally the immediate surroundings. As this division depends on the location of theboundary there is some arbitrariness in the classification (by locating the boundary to take in the© 1999 by CRC Press LLC2-8Section 2immediate surroundings, all irreversibilities are internal). Nonetheless, valuable insights can result whenthis distinction between irreversibilities is made.

When internal irreversibilities are absent during aprocess, the process is said to be internally reversible. At every intermediate state of an internallyreversible process of a closed system, all intensive properties are uniform throughout each phase present:the temperature, pressure, specific volume, and other intensive properties do not vary with position.

Thediscussions to follow compare the actual and internally reversible process concepts for two cases ofspecial interest.For a gas as the system, the work of expansion arises from the force exerted by the system to movethe boundary against the resistance offered by the surroundings:W=∫2Fdx =1∫2pAdx1where the force is the product of the moving area and the pressure exerted by the system there. Notingthat Adx is the change in total volume of the system,W=∫2pdV1This expression for work applies to both actual and internally reversible expansion processes. However,for an internally reversible process p is not only the pressure at the moving boundary but also the pressureof the entire system. Furthermore, for an internally reversible process the volume equals mv, where thespecific volume v has a single value throughout the system at a given instant. Accordingly, the work ofan internally reversible expansion (or compression) process isW=m∫ pdv2(2.10)1When such a process of a closed system is represented by a continuous curve on a plot of pressure vs.specific volume, the area under the curve is the magnitude of the work per unit of system mass (areaa-b-c′-d′ of Figure 2.3, for example).Although improved thermodynamic performance can accompany the reduction of irreversibilities,steps in this direction are normally constrained by a number of practical factors often related to costs.For example, consider two bodies able to communicate thermally.

With a finite temperature differencebetween them, a spontaneous heat transfer would take place and, as noted previously, this would be asource of irreversibility. The importance of the heat transfer irreversibility diminishes as the temperaturedifference narrows; and as the temperature difference between the bodies vanishes, the heat transferapproaches ideality.

From the study of heat transfer it is known, however, that the transfer of a finiteamount of energy by heat between bodies whose temperatures differ only slightly requires a considerableamount of time, a large heat transfer surface area, or both. To approach ideality, therefore, a heat transferwould require an exceptionally long time and/or an exceptionally large area, each of which has costimplications constraining what can be achieved practically.Carnot CorollariesThe two corollaries of the second law known as Carnot corollaries state: (1) the thermal efficiency ofan irreversible power cycle is always less than the thermal efficiency of a reversible power cycle wheneach operates between the same two thermal reservoirs; (2) all reversible power cycles operating betweenthe same two thermal reservoirs have the same thermal efficiency.

A cycle is considered reversible whenthere are no irreversibilities within the system as it undergoes the cycle, and heat transfers between thesystem and reservoirs occur ideally (that is, with a vanishingly small temperature difference).© 1999 by CRC Press LLC2-9Engineering ThermodynamicsKelvin Temperature ScaleCarnot corollary 2 suggests that the thermal efficiency of a reversible power cycle operating betweentwo thermal reservoirs depends only on the temperatures of the reservoirs and not on the nature of thesubstance making up the system executing the cycle or the series of processes. With Equation 2.9 it canbe concluded that the ratio of the heat transfers is also related only to the temperatures, and is independentof the substance and processes: QC  Q  rev = ψ (TC , TH ) Hcyclewhere QH is the energy transferred to the system by heat transfer from a hot reservoir at temperatureTH, and QC is the energy rejected from the system to a cold reservoir at temperature TC.

The words revcycle emphasize that this expression applies only to systems undergoing reversible cycles while operatingbetween the two reservoirs. Alternative temperature scales correspond to alternative specifications forthe function ψ in this relation.The Kelvin temperature scale is based on ψ(TC, TH) = TC /TH. Then QC TC Q  rev = T HH(2.11)cycleThis equation defines only a ratio of temperatures. The specification of the Kelvin scale is completedby assigning a numerical value to one standard reference state.

The state selected is the same used todefine the gas scale: at the triple point of water the temperature is specified to be 273.16 K. If a reversiblecycle is operated between a reservoir at the reference-state temperature and another reservoir at anunknown temperature T, then the latter temperature is related to the value at the reference state by QT = 273.16  Q ′  revcyclewhere Q is the energy received by heat transfer from the reservoir at temperature T, and Q′ is the energyrejected to the reservoir at the reference temperature. Accordingly, a temperature scale is defined that isvalid over all ranges of temperature and that is independent of the thermometric substance.Carnot EfficiencyFor the special case of a reversible power cycle operating between thermal reservoirs at temperaturesTH and TC on the Kelvin scale, combination of Equations 2.9 and 2.11 results inηmax = 1 −TCTH(2.12)called the Carnot efficiency.

This is the efficiency of all reversible power cycles operating betweenthermal reservoirs at TH and TC. Moreover, it is the maximum theoretical efficiency that any power cycle,real or ideal, could have while operating between the same two reservoirs. As temperatures on theRankine scale differ from Kelvin temperatures only by the factor 1.8, the above equation may be appliedwith either scale of temperature.© 1999 by CRC Press LLC2-10Section 2The Clausius InequalityThe Clausius inequality provides the basis for introducing two ideas instrumental for quantitativeevaluations of processes of systems from a second law perspective: entropy and entropy generation.

TheClausius inequality states that δQ ∫  T ≤0(2.13a)bwhere δQ represents the heat transfer at a part of the system boundary during a portion of the cycle,and T is the absolute temperature at that part of the boundary. The symbol δ is used to distinguish thedifferentials of nonproperties, such as heat and work, from the differentials of properties, written withthe symbol d. The subscript b indicates that the integrand is evaluated at the boundary of the systemexecuting the cycle. The symbol ∫ indicates that the integral is to be performed over all parts of theboundary and over the entire cycle.

The Clausius inequality can be demonstrated using the Kelvin-Planckstatement of the second law, and the significance of the inequality is the same: the equality applies whenthere are no internal irreversibilities as the system executes the cycle, and the inequality applies wheninternal irreversibilities are present.The Clausius inequality can be expressed alternatively as δQ ∫  T = − Sgen(2.13b)bwhere Sgen can be viewed as representing the strength of the inequality. The value of Sgen is positivewhen internal irreversibilities are present, zero when no internal irreversibilities are present, and cannever be negative.

Accordingly, Sgen is a measure of the irreversibilities present within the systemexecuting the cycle. In the next section, Sgen is identified as the entropy generated (or produced) byinternal irreversibilities during the cycle.Entropy and Entropy GenerationEntropyConsider two cycles executed by a closed system.

One cycle consists of an internally reversible processA from state 1 to state 2, followed by an internally reversible process C from state 2 to state 1. Theother cycle consists of an internally reversible process B from state 1 to state 2, followed by the sameprocess C from state 2 to state 1 as in the first cycle. For these cycles, Equation 2.13b takes the form∫11δQ  +T A 2δQ  = − Sgen = 0T C∫δQ  +T B ∫1δQ  = − Sgen = 0T C∫2212where Sgen has been set to zero since the cycles are composed of internally reversible processes.Subtracting these equations leaves© 1999 by CRC Press LLC∫21δQ  =T A ∫21δQ T B2-11Engineering ThermodynamicsSince A and B are arbitrary, it follows that the integral of δQ/T has the same value for any internallyreversible process between the two states: the value of the integral depends on the end states only.

Itcan be concluded, therefore, that the integral defines the change in some property of the system. Selectingthe symbol S to denote this property, its change is given byS2 − S1 = ∫δQ T  int21(2.14a)revwhere the subscript int rev indicates that the integration is carried out for any internally reversible processlinking the two states.

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