The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 4
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This extensive property is called entropy.Since entropy is a property, the change in entropy of a system in going from one state to another isthe same for all processes, both internally reversible and irreversible, between these two states. In otherwords, once the change in entropy between two states has been evaluated, this is the magnitude of theentropy change for any process of the system between these end states.The definition of entropy change expressed on a differential basis is δQ dS = T int(2.14b)revEquation 2.14b indicates that when a closed system undergoing an internally reversible process receivesenergy by heat transfer, the system experiences an increase in entropy. Conversely, when energy isremoved from the system by heat transfer, the entropy of the system decreases.
This can be interpretedto mean that an entropy transfer is associated with (or accompanies) heat transfer. The direction of theentropy transfer is the same as that of the heat transfer. In an adiabatic internally reversible process ofa closed system the entropy would remain constant. A constant entropy process is called an isentropicprocess.On rearrangement, Equation 2.14b becomes(δQ)intrev = TdSThen, for an internally reversible process of a closed system between state 1 and state 2,∫2Qint = m Tds(2.15)1revWhen such a process is represented by a continuous curve on a plot of temperature vs. specific entropy,the area under the curve is the magnitude of the heat transfer per unit of system mass.Entropy BalanceFor a cycle consisting of an actual process from state 1 to state 2, during which internal irreversibilitiesare present, followed by an internally reversible process from state 2 to state 1, Equation 2.13b takesthe form∫21 δQ + T b δQ ∫ T 12intrev= − Sgenwhere the first integral is for the actual process and the second integral is for the internally reversibleprocess.
Since no irreversibilities are associated with the internally reversible process, the term Sgenaccounting for the effect of irreversibilities during the cycle can be identified with the actual process only.© 1999 by CRC Press LLC2-12Section 2Applying the definition of entropy change, the second integral of the foregoing equation can beexpressed asS1 − S2 = δQ ∫ T 12intrevIntroducing this and rearranging the equation, the closed system entropy balance results:S2 − S1 = δQ ∫ T 21+ Sgenb______ ____________entropy entropychange transferentropygeneration(2.16)When the end states are fixed, the entropy change on the left side of Equation 2.16 can be evaluatedindependently of the details of the process from state 1 to state 2.
However, the two terms on the rightside depend explicitly on the nature of the process and cannot be determined solely from knowledge ofthe end states. The first term on the right side is associated with heat transfer to or from the systemduring the process. This term can be interpreted as the entropy transfer associated with (or accompanying)heat transfer. The direction of entropy transfer is the same as the direction of the heat transfer, and thesame sign convention applies as for heat transfer: a positive value means that entropy is transferred intothe system, and a negative value means that entropy is transferred out.The entropy change of a system is not accounted for solely by entropy transfer, but is also due to thesecond term on the right side of Equation 2.16 denoted by Sgen. The term Sgen is positive when internalirreversibilities are present during the process and vanishes when internal irreversibilities are absent.This can be described by saying that entropy is generated (or produced) within the system by the actionof irreversibilities.
The second law of thermodynamics can be interpreted as specifying that entropy isgenerated by irreversibilities and conserved only in the limit as irreversibilities are reduced to zero. SinceSgen measures the effect of irreversibilities present within a system during a process, its value dependson the nature of the process and not solely on the end states.
Entropy generation is not a property.When applying the entropy balance, the objective is often to evaluate the entropy generation term.However, the value of the entropy generation for a given process of a system usually does not havemuch significance by itself. The significance is normally determined through comparison. For example,the entropy generation within a given component might be compared to the entropy generation valuesof the other components included in an overall system formed by these components. By comparingentropy generation values, the components where appreciable irreversibilities occur can be identifiedand rank ordered. This allows attention to be focused on the components that contribute most heavilyto inefficient operation of the overall system.To evaluate the entropy transfer term of the entropy balance requires information regarding both theheat transfer and the temperature on the boundary where the heat transfer occurs.
The entropy transferterm is not always subject to direct evaluation, however, because the required information is eitherunknown or undefined, such as when the system passes through states sufficiently far from equilibrium.In practical applications, it is often convenient, therefore, to enlarge the system to include enough ofthe immediate surroundings that the temperature on the boundary of the enlarged system correspondsto the ambient temperature, Tamb. The entropy transfer term is then simply Q/Tamb.
However, as theirreversibilities present would not be just those for the system of interest but those for the enlargedsystem, the entropy generation term would account for the effects of internal irreversibilities within the© 1999 by CRC Press LLC2-13Engineering Thermodynamicssystem and external irreversibilities present within that portion of the surroundings included within theenlarged system.A form of the entropy balance convenient for particular analyses is the rate form:dS=dtQ˙ j∑Tj+ S˙gen(2.17)jwhere dS/dt is the time rate of change of entropy of the system. The term Q˙ j / Tj represents the timerate of entropy transfer through the portion of the boundary whose instantaneous temperature is Tj.
Theterm Ṡgen accounts for the time rate of entropy generation due to irreversibilities within the system.For a system isolated from its surroundings, the entropy balance is(S2 − S1 )isol = Sgen(2.18)where Sgen is the total amount of entropy generated within the isolated system. Since entropy is generatedin all actual processes, the only processes of an isolated system that actually can occur are those forwhich the entropy of the isolated system increases. This is known as the increase of entropy principle.© 1999 by CRC Press LLC2-14Section 22.2 Control Volume ApplicationsSince most applications of engineering thermodynamics are conducted on a control volume basis, thecontrol volume formulations of the mass, energy, and entropy balances presented in this section areespecially important.
These are given here in the form of overall balances. Equations of change formass, energy, and entropy in the form of differential equations are also available in the literature (see,e.g., Bird et al., 1960).Conservation of MassWhen applied to a control volume, the principle of mass conservation states: The time rate of accumulation of mass within the control volume equals the difference between the total rates of mass flow inand out across the boundary.
An important case for engineering practice is one for which inward andoutward flows occur, each through one or more ports. For this case the conservation of mass principletakes the formdmcv=dt∑ m˙ − ∑ m˙ii(2.19)eeThe left side of this equation represents the time rate of change of mass contained within the controlvolume, ṁi denotes the mass flow rate at an inlet, and ṁe is the mass flow rate at an outlet.The volumetric flow rate through a portion of the control surface with area dA is the product of thevelocity component normal to the area, vn, times the area: vn dA.
The mass flow rate through dA is ρ(vndA). The mass rate of flow through a port of area A is then found by integration over the areaṁ =∫ ρvAndAFor one-dimensional flow the intensive properties are uniform with position over area A, and the lastequation becomesṁ = ρvA =vAv(2.20)where v denotes the specific volume and the subscript n has been dropped from velocity for simplicity.Control Volume Energy BalanceWhen applied to a control volume, the principle of energy conservation states: The time rate of accumulation of energy within the control volume equals the difference between the total incoming rate ofenergy transfer and the total outgoing rate of energy transfer.
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