Moran, M. J., Tsatsaronis, G. Engineering Thermodynamics. The CRC Handbook of Thermal Engineering. Ed. Frank Kreith Boca Raton: CRC Press LLC, 2000

Transcription

1 Moran, M. J., satsarons, G. Engneerng hermodynamcs. he CRC Handbook of hermal Engneerng. Ed. Frank Kreth Boca Raton: CRC Press LLC, 000

2 Engneerng hermodynamcs Mchael J. Moran he Oho State Unersty George satsarons echnsche Unerstät Berln. Fundamentals Basc Concets and Defntons he Frst Law of hermodynamcs, Energy he Second Law of hermodynamcs, Entroy Entroy and Entroy Generaton. Control Volume Alcatons Conseraton of Mass Control Volume Energy Balance Control Volume Entroy Balance Control Volumes at Steady State.3 Proerty Relatons and Data Basc Relatons for Pure Substances P-- Relatons Ealuatng h, u, and s Fundamental hermodynamc Functons hermodynamc Data Retreal Ideal Gas Model Generalzed Charts for Enthaly, Entroy, and Fugacty Multcomonent Systems.4 Combuston Reacton Equatons Proerty Data for Reacte Systems Reacton Equlbrum.5 Exergy Analyss Defnng Exergy Control Volume Exergy Rate Balance Exergetc Effcency Introducton to Exergy Costng.6 Vaor and Gas Power Cycles Rankne and Brayton Cycles Otto, Desel, and Dual Cycles Carnot, Ercsson, and Strlng Cycles.7 Gudelnes for Imrong hermodynamc Effecteness.8 Exergoeconomcs Exergy Costng Cost Balance Auxlary Costng Equatons General Examle Exergoeconomc Varables and Ealuaton.9 Desgn Otmzaton An Iterate Exergoeconomc Procedure for Otmzng the Desgn of a hermal System Case Study Addtonal Iteratons.0 Economc Analyss of hermal Systems Estmaton of otal Catal Inestment Prncles of Economc Ealuaton Calculaton of the Product Costs Although arous asects of what s now known as thermodynamcs hae been of nterest snce antquty, formal study began only n the early 9th century through consderaton of the mote ower of heat: the caacty of hot bodes to roduce work. oday the scoe s larger, dealng generally wth energy and

3 entroy, and wth relatonshs among the roertes of matter. Moreoer, n the ast 5 years engneerng thermodynamcs has undergone a reoluton, both n terms of the resentaton of fundamentals and n the manner that t s aled. In artcular, the second law of thermodynamcs has emerged as an effecte tool for engneerng analyss and desgn.. Fundamentals Classcal thermodynamcs s concerned rmarly wth the macrostructure of matter. It addresses the gross characterstcs of large aggregatons of molecules and not the behaor of nddual molecules. he mcrostructure of matter s studed n knetc theory and statstcal mechancs (ncludng quantum thermodynamcs). In ths chater, the classcal aroach to thermodynamcs s featured. Basc Concets and Defntons hermodynamcs s both a branch of hyscs and an engneerng scence. he scentst s normally nterested n ganng a fundamental understandng of the hyscal and chemcal behaor of fxed, quescent quanttes of matter and uses the rncles of thermodynamcs to relate the roertes of matter. Engneers are generally nterested n studyng systems and how they nteract wth ther surroundngs. o facltate ths, engneers hae extended the subect of thermodynamcs to the study of systems through whch matter flows. System In a thermodynamc analyss, the system s the subect of the nestgaton. Normally the system s a secfed quantty of matter and/or a regon that can be searated from eerythng else by a well-defned surface. he defnng surface s known as the control surface or system boundary. he control surface may be moable or fxed. Eerythng external to the system s the surroundngs. A system of fxed mass s referred to as a control mass or as a closed system. When there s flow of mass through the control surface, the system s called a control olume, or oen, system. An solated system s a closed system that does not nteract n any way wth ts surroundngs. State, Proerty he condton of a system at any nstant of tme s called ts state. he state at a gen nstant of tme s descrbed by the roertes of the system. A roerty s any quantty whose numercal alue deends on the state but not the hstory of the system. he alue of a roerty s determned n rncle by some tye of hyscal oeraton or test. Extense roertes deend on the sze or extent of the system. Volume, mass, energy, and entroy are examles of extense roertes. An extense roerty s addte n the sense that ts alue for the whole system equals the sum of the alues for ts arts. Intense roertes are ndeendent of the sze or extent of the system. Pressure and temerature are examles of ntense roertes. A mole s a quantty of substance hang a mass numercally equal to ts molecular weght. Desgnatng the molecular weght by M and the number of moles by n, the mass m of the substance s m = nm. One klogram mole, desgnated kmol, of oxygen s 3.0 kg and one ound mole (lbmol) s 3.0 lb. When an extense roerty s reorted on a unt mass or a unt mole bass, t s called a secfc roerty. An oerbar s used to dstngush an extense roerty wrtten on a er-mole bass from ts alue exressed er unt mass. For examle, the olume er mole s, whereas the olume er unt mass s, and the two secfc olumes are related by = M. Process, Cycle wo states are dentcal f, and only f, the roertes of the two states are dentcal. When any roerty of a system changes n alue there s a change n state, and the system s sad to undergo a rocess. When a system n a gen ntal state goes through a sequence of rocesses and fnally returns to ts ntal state, t s sad to hae undergone a cycle.

4 Phase and Pure Substance he term hase refers to a quantty of matter that s homogeneous throughout n both chemcal comoston and hyscal structure. Homogenety n hyscal structure means that the matter s all sold, or all lqud, or all aor (or, equalently, all gas). A system can contan one or more hases. For examle, a system of lqud water and water aor (steam) contans two hases. A ure substance s one that s unform and narable n chemcal comoston. A ure substance can exst n more than one hase, but ts chemcal comoston must be the same n each hase. For examle, f lqud water and water aor form a system wth two hases, the system can be regarded as a ure substance because each hase has the same comoston. he nature of hases that coexst n equlbrum s addressed by the hase rule (Secton.3, Multcomonent Systems). Equlbrum Equlbrum means a condton of balance. In thermodynamcs the concet ncludes not only a balance of forces, but also a balance of other nfluences. Each knd of nfluence refers to a artcular asect of thermodynamc (comlete) equlbrum. hermal equlbrum refers to an equalty of temerature, mechancal equlbrum to an equalty of ressure, and hase equlbrum to an equalty of chemcal otentals (Secton.3, Multcomonent Systems). Chemcal equlbrum s also establshed n terms of chemcal otentals (Secton.4, Reacton Equlbrum). For comlete equlbrum, the seeral tyes of equlbrum must exst nddually. o determne f a system s n thermodynamc equlbrum, one may thnk of testng t as follows: solate the system from ts surroundngs and watch for changes n ts obserable roertes. If there are no changes, t may be concluded that the system was n equlbrum at the moment t was solated. he system can be sad to be at an equlbrum state. When a system s solated, t cannot nteract wth ts surroundngs; howeer, ts state can change as a consequence of sontaneous eents occurrng nternally as ts ntense roertes, such as temerature and ressure, tend toward unform alues. When all such changes cease, the system s n equlbrum. At equlbrum. temerature and ressure are unform throughout. If graty s sgnfcant, a ressure araton wth heght can exst, as n a ertcal column of lqud. emerature A scale of temerature ndeendent of the thermometrc substance s called a thermodynamc temerature scale. he Keln scale, a thermodynamc scale, can be elcted from the second law of thermodynamcs (Secton., he Second Law of hermodynamcs, Entroy). he defnton of temerature followng from the second law s ald oer all temerature ranges and rodes an essental connecton between the seeral emrcal measures of temerature. In artcular, temeratures ealuated usng a constantolume gas thermometer are dentcal to those of the Keln scale oer the range of temeratures where gas thermometry can be used. he emrcal gas scale s based on the exermental obseratons that () at a gen temerature leel all gases exhbt the same alue of the roduct ( s ressure and the secfc olume on a molar bass) f the ressure s low enough, and () the alue of the roduct ncreases wth the temerature leel. On ths bass the gas temerature scale s defned by lm R 0 = where s temerature and R s the unersal gas constant. he absolute temerature at the trle ont of water (Secton.3, P-- Relatons) s fxed by nternatonal agreement to be 73.6 K on the Keln temerature scale. R s then ealuated exermentally as R = 8.34 kj/kmol K (545 ft lbf/lbmol R). he Celsus termerature scale (also called the centgrade scale) uses the degree Celsus ( C), whch has the same magntude as the Keln. hus, temerature dfferences are dentcal on both scales. Howeer, the zero ont on the Celsus scale s shfted to 73.5 K, as shown by the followng relatonsh between the Celsus temerature and the Keln temerature:

5 = C K (.) On the Celsus scale, the trle ont of water s 0.0 C and 0 K corresonds to 73.5 C. wo other temerature scales are commonly used n engneerng n the U.S. By defnton, the Rankne scale, the unt of whch s the degree Rankne ( R), s roortonal to the Keln temerature accordng to ( R)= 8. ( K) (.) he Rankne scale s also an absolute thermodynamc scale wth an absolute zero that concdes wth the absolute zero of the Keln scale. In thermodynamc relatonshs, temerature s always n terms of the Keln or Rankne scale unless secfcally stated otherwse. A degree of the same sze as that on the Rankne scale s used n the Fahrenhet scale, but the zero ont s shfted accordng to the relaton ( F)= ( R) (.3) Substtutng Equatons. and. nto Equaton.3 ges ( F)= 8. ( C)+ 3 (.4) hs equaton shows that the Fahrenhet temerature of the ce ont (0 C) s 3 F and of the steam ont (00 C) s F. he 00 Celsus or Keln degrees between the ce ont and steam ont corresonds to 80 Fahrenhet or Rankne degrees. o rode a standard for temerature measurement takng nto account both theoretcal and ractcal consderatons, the Internatonal emerature Scale of 990 (IS-90) s defned n such a way that the temerature measured on t conforms wth the thermodynamc temerature, the unt of whch s the Keln, to wthn the lmts of accuracy of measurement obtanable n 990. Further dscusson of IS- 90 s roded by Preston-homas (990). he Frst Law of hermodynamcs, Energy Energy s a fundamental concet of thermodynamcs and one of the most sgnfcant asects of engneerng analyss. Energy can be stored wthn systems n arous macroscoc forms: knetc energy, gratatonal otental energy, and nternal energy. Energy can also be transformed from one form to another and transferred between systems. For closed systems, energy can be transferred by work and heat transfer. he total amount of energy s consered n all transformatons and transfers. Work In thermodynamcs, the term work denotes a means for transferrng energy. Work s an effect of one system on another that s dentfed and measured as follows: work s done by a system on ts surroundngs f the sole effect on eerythng external to the system could hae been the rasng of a weght. he test of whether a work nteracton has taken lace s not that the eleaton of a weght s actually changed, nor that a force actually acted through a dstance, but that the sole effect could be the change n eleaton of a weght. he magntude of the work s measured by the number of standard weghts that could hae been rased. Snce the rasng of a weght s n effect a force actng through a dstance, the work concet of mechancs s resered. hs defnton ncludes work effects such as s assocated wth rotatng shafts, dslacement of the boundary, and the flow of electrcty. Work done by a system s consdered oste: W > 0. Work done on a system s consdered negate: W < 0. he tme rate of dong work, or ower, s symbolzed by Ẇ and adheres to the same sgn conenton.

6 Energy A closed system undergong a rocess that noles only work nteractons wth ts surroundngs exerences an adabatc rocess. On the bass of exermental edence, t can be ostulated that when a closed system s altered adabatcally, the amount of work s fxed by the end states of the system and s ndeendent of the detals of the rocess. hs ostulate, whch s one way the frst law of thermodynamcs can be stated, can be made regardless of the tye of work nteracton noled, the tye of rocess, or the nature of the system. As the work n an adabatc rocess of a closed system s fxed by the end states, an extense roerty called energy can be defned for the system such that ts change between two states s the work n an adabatc rocess that has these as the end states. In engneerng thermodynamcs the change n the energy of a system s consdered to be made u of three macroscoc contrbutons: the change n knetc energy, KE, assocated wth the moton of the system as a whole relate to an external coordnate frame, the change n gratatonal otental energy, PE, assocated wth the oston of the system as a whole n the Earth s gratatonal feld, and the change n nternal energy, U, whch accounts for all other energy assocated wth the system. Lke knetc energy and gratatonal otental energy, nternal energy s an extense roerty. In summary, the change n energy between two states of a closed system n terms of the work W ad of an adabatc rocess between these states s ( KE KE )+( PE PE )+( U U )= W ad (.5) where and denote the ntal and fnal states, resectely, and the mnus sgn before the work term s n accordance wth the reously stated sgn conenton for work. Snce any arbtrary alue can be assgned to the energy of a system at a gen state, no artcular sgnfcance can be attached to the alue of the energy at state or at any other state. Only changes n the energy of a system hae sgnfcance. he secfc energy (energy er unt mass) s the sum of the secfc nternal energy, u, the secfc knetc energy, /, and the secfc gratatonal otental energy, gz, such that secfc energy = u + +gz (.6) where the elocty and the eleaton z are each relate to secfed datums (often the Earth s surface) and g s the acceleraton of graty. A roerty related to nternal energy u, ressure, and secfc olume s enthaly, defned by or on an extense bass h = u+ H = U + V (.7a) (.7b) Heat Closed systems can also nteract wth ther surroundngs n a way that cannot be categorzed as work, as, for examle, a gas (or lqud) contaned n a closed essel undergong a rocess whle n contact wth a flame. hs tye of nteracton s called a heat nteracton, and the rocess s referred to as nonadabatc. A fundamental asect of the energy concet s that energy s consered. hus, snce a closed system exerences recsely the same energy change durng a nonadabatc rocess as durng an adabatc

7 rocess between the same end states, t can be concluded that the net energy transfer to the system n each of these rocesses must be the same. It follows that heat nteractons also nole energy transfer. Denotng the amount of energy transferred to a closed system n heat nteractons by Q, these consderatons can be summarzed by the closed system energy balance: ( U U )+( KE KE )+( PE PE )= Q W (.8) he closed system energy balance exresses the conseraton of energy rncle for closed systems of all knds. he quantty denoted by Q n Equaton.8 accounts for the amount of energy transferred to a closed system durng a rocess by means other than work. On the bass of an exerment, t s known that such an energy transfer s nduced only as a result of a temerature dfference between the system and ts surroundngs and occurs only n the drecton of decreasng temerature. hs means of energy transfer s called an energy transfer by heat. he followng sgn conenton ales: Q > 0: Q < 0: heat transfer tothe system heat transfer from the system he tme rate of heat transfer, denoted by Q, adheres to the same sgn conenton. Methods based on exerment are aalable for ealuatng energy transfer by heat. hese methods recognze two basc transfer mechansms: conducton and thermal radaton. In addton, theoretcal and emrcal relatonshs are aalable for ealuatng energy transfer nolng combned modes such as conecton. Further dscusson of heat transfer fundamentals s roded n Chater 3. he quanttes symbolzed by W and Q account for transfers of energy. he terms work and heat denote dfferent means whereby energy s transferred and not what s transferred. Work and heat are not roertes, and t s mroer to seak of work or heat contaned n a system. Howeer, to achee economy of exresson n subsequent dscussons, W and Q are often referred to smly as work and heat transfer, resectely. hs less formal aroach s commonly used n engneerng ractce. Power Cycles Snce energy s a roerty, oer each cycle there s no net change n energy. hus, Equaton.8 reads for any cycle hat s, for any cycle the net amount of energy receed through heat nteractons s equal to the net energy transferred out n work nteractons. A ower cycle, or heat engne, s one for whch a net amount of energy s transferred out by work: W cycle > 0. hs equals the net amount of energy transferred n by heat. Power cycles are characterzed both by addton of energy by heat transfer, Q A, and netable reectons of energy by heat transfer, Q R : Combnng the last two equatons, Q cycle = W cycle Qcycle = QA QR Wcycle = QA QR he thermal effcency of a heat engne s defned as the rato of the net work deeloed to the total energy added by heat transfer:

8 Wcycle Q η= = Q Q A R A (.9) he thermal effcency s strctly less than 00%. hat s, some orton of the energy Q A suled s narably reected Q R 0. he Second Law of hermodynamcs, Entroy Many statements of the second law of thermodynamcs hae been roosed. Each of these can be called a statement of the second law or a corollary of the second law snce, f one s nald, all are nald. In eery nstance where a consequence of the second law has been tested drectly or ndrectly by exerment t has been erfed. Accordngly, the bass of the second law, lke eery other hyscal law, s exermental edence. Keln-Planck Statement he Keln-Plank statement of the second law of thermodynamcs refers to a thermal reseror. A thermal reseror s a system that remans at a constant temerature een though energy s added or remoed by heat transfer. A reseror s an dealzaton, of course, but such a system can be aroxmated n a number of ways by the Earth s atmoshere, large bodes of water (lakes, oceans), and so on. Extense roertes of thermal reserors, such as nternal energy, can change n nteractons wth other systems een though the reseror temerature remans constant, howeer. he Keln-Planck statement of the second law can be gen as follows: It s mossble for any system to oerate n a thermodynamc cycle and deler a net amount of energy by work to ts surroundngs whle receng energy by heat transfer from a sngle thermal reseror. In other words, a eretualmoton machne of the second knd s mossble. Exressed analytcally, the Keln-Planck statement s W cycle 0 ( sngle reseror) where the words sngle reseror emhasze that the system communcates thermally only wth a sngle reseror as t executes the cycle. he less than sgn ales when nternal rreersbltes are resent as the system of nterest undergoes a cycle and the equal to sgn ales only when no rreersbltes are resent. Irreersbltes A rocess s sad to be reersble f t s ossble for ts effects to be eradcated n the sense that there s some way by whch both the system and ts surroundngs can be exactly restored to ther resecte ntal states. A rocess s rreersble f there s no way to undo t. hat s, there s no means by whch the system and ts surroundngs can be exactly restored to ther resecte ntal states. A system that has undergone an rreersble rocess s not necessarly recluded from beng restored to ts ntal state. Howeer, were the system restored to ts ntal state, t would not also be ossble to return the surroundngs to ther ntal state. here are many effects whose resence durng a rocess renders t rreersble. hese nclude, but are not lmted to, the followng: heat transfer through a fnte temerature dfference; unrestraned exanson of a gas or lqud to a lower ressure; sontaneous chemcal reacton; mxng of matter at dfferent comostons or states; frcton (sldng frcton as well as frcton n the flow of fluds); electrc current flow through a resstance; magnetzaton or olarzaton wth hysteress; and nelastc deformaton. he term rreersblty s used to dentfy effects such as these. Irreersbltes can be dded nto two classes, nternal and external. Internal rreersbltes are those that occur wthn the system, whle external rreersbltes are those that occur wthn the surroundngs, normally the mmedate surroundngs. As ths dson deends on the locaton of the boundary there s some arbtrarness n the classfcaton (by locatng the boundary to take n the

9 mmedate surroundngs, all rreersbltes are nternal). Nonetheless, aluable nsghts can result when ths dstncton between rreersbltes s made. When nternal rreersbltes are absent durng a rocess, the rocess s sad to be nternally reersble. At eery ntermedate state of an nternally reersble rocess of a closed system, all ntense roertes are unform throughout each hase resent: the temerature, ressure, secfc olume, and other ntense roertes do not ary wth oston. he dscussons to follow comare the actual and nternally reersble rocess concets for two cases of secal nterest. For a gas as the system, the work of exanson arses from the force exerted by the system to moe the boundary aganst the resstance offered by the surroundngs: W = Fdx = Adx where the force s the roduct of the mong area and the ressure exerted by the system there. Notng that Adx s the change n total olume of the system, W hs exresson for work ales to both actual and nternally reersble exanson rocesses. Howeer, for an nternally reersble rocess s not only the ressure at the mong boundary but also the ressure of the entre system. Furthermore, for an nternally reersble rocess the olume equals m, where the secfc olume has a sngle alue throughout the system at a gen nstant. Accordngly, the work of an nternally reersble exanson (or comresson) rocess s = dv W = m d (.0) When such a rocess of a closed system s reresented by a contnuous cure on a lot of ressure s. secfc olume, the area under the cure s the magntude of the work er unt of system mass (area a-b-c -d of Fgure.3, for examle). Although mroed thermodynamc erformance can accomany the reducton of rreersbltes, stes n ths drecton are normally constraned by a number of ractcal factors often related to costs. For examle, consder two bodes able to communcate thermally. Wth a fnte temerature dfference between them, a sontaneous heat transfer would take lace and, as noted reously, ths would be a source of rreersblty. he mortance of the heat transfer rreersblty dmnshes as the temerature dfference narrows; and as the temerature dfference between the bodes anshes, the heat transfer aroaches dealty. From the study of heat transfer t s known, howeer, that the transfer of a fnte amount of energy by heat between bodes whose temeratures dffer only slghtly requres a consderable amount of tme, a large heat transfer surface area, or both. o aroach dealty, therefore, a heat transfer would requre an excetonally long tme and/or an excetonally large area, each of whch has cost mlcatons constranng what can be acheed ractcally. Carnot Corollares he two corollares of the second law known as Carnot corollares state: () the thermal effcency of an rreersble ower cycle s always less than the thermal effcency of a reersble ower cycle when each oerates between the same two thermal reserors; () all reersble ower cycles oeratng between the same two thermal reserors hae the same thermal effcency. A cycle s consdered reersble when there are no rreersbltes wthn the system as t undergoes the cycle, and heat transfers between the system and reserors occur deally (that s, wth a anshngly small temerature dfference).

10 Keln emerature Scale Carnot corollary suggests that the thermal effcency of a reersble ower cycle oeratng between two thermal reserors deends only on the temeratures of the reserors and not on the nature of the substance makng u the system executng the cycle or the seres of rocesses. Wth Equaton.9 t can be concluded that the rato of the heat transfers s also related only to the temeratures, and s ndeendent of the substance and rocesses: Q Q C H = ψ ( re C, H) cycle where Q H s the energy transferred to the system by heat transfer from a hot reseror at temerature H, and Q C s the energy reected from the system to a cold reseror at temerature C. he words re cycle emhasze that ths exresson ales only to systems undergong reersble cycles whle oeratng between the two reserors. Alternate temerature scales corresond to alternate secfcatons for the functon ψ n ths relaton. he Keln temerature scale s based on ψ( C, H ) = C / H. hen Q Q C H = re cycle C H (.) hs equaton defnes only a rato of temeratures. he secfcaton of the Keln scale s comleted by assgnng a numercal alue to one standard reference state. he state selected s the same used to defne the gas scale: at the trle ont of water the temerature s secfed to be 73.6 K. If a reersble cycle s oerated between a reseror at the reference-state temerature and another reseror at an unknown temerature, then the latter temerature s related to the alue at the reference state by Q = Q where Q s the energy receed by heat transfer from the reseror at temerature, and Q s the energy reected to the reseror at the reference temerature. Accordngly, a temerature scale s defned that s ald oer all ranges of temerature and that s ndeendent of the thermometrc substance. Carnot Effcency For the secal case of a reersble ower cycle oeratng between thermal reserors at temeratures H and C on the Keln scale, combnaton of Equatons.9 and. results n re cycle η max = C H (.) called the Carnot effcency. hs s the effcency of all reersble ower cycles oeratng between thermal reserors at H and C. Moreoer, t s the maxmum theoretcal effcency that any ower cycle, real or deal, could hae whle oeratng between the same two reserors. As temeratures on the Rankne scale dffer from Keln temeratures only by the factor.8, the aboe equaton may be aled wth ether scale of temerature.

11 he Clausus Inequalty he Clausus nequalty rodes the bass for ntroducng two deas nstrumental for quanttate ealuatons of rocesses of systems from a second law ersecte: entroy and entroy generaton. he Clausus nequalty states that δq 0 b (.3a) where δq reresents the heat transfer at a art of the system boundary durng a orton of the cycle, and s the absolute temerature at that art of the boundary. he symbol δ s used to dstngush the dfferentals of nonroertes, such as heat and work, from the dfferentals of roertes, wrtten wth the symbol d. he subscrt b ndcates that the ntegrand s ealuated at the boundary of the system executng the cycle. he symbol ndcates that the ntegral s to be erformed oer all arts of the boundary and oer the entre cycle. he Clausus nequalty can be demonstrated usng the Keln-Planck statement of the second law, and the sgnfcance of the nequalty s the same: the equalty ales when there are no nternal rreersbltes as the system executes the cycle, and the nequalty ales when nternal rreersbltes are resent. he Clausus nequalty can be exressed alternately as δq = S b gen (.3b) where S gen can be ewed as reresentng the strength of the nequalty. he alue of S gen s oste when nternal rreersbltes are resent, zero when no nternal rreersbltes are resent, and can neer be negate. Accordngly, S gen s a measure of the rreersbltes resent wthn the system executng the cycle. In the next secton, S gen s dentfed as the entroy generated (or roduced) by nternal rreersbltes durng the cycle. Entroy and Entroy Generaton Entroy Consder two cycles executed by a closed system. One cycle conssts of an nternally reersble rocess A from state to state, followed by an nternally reersble rocess C from state to state. he other cycle conssts of an nternally reersble rocess B from state to state, followed by the same rocess C from state to state as n the frst cycle. For these cycles, Equaton.3b takes the form δq δq + Sgen 0 = = A δq δq + Sgen 0 = = B where S gen has been set to zero snce the cycles are comosed of nternally reersble rocesses. Subtractng these equatons leaes C C δq δq = A B

12 Snce A and B are arbtrary, t follows that the ntegral of δq/ has the same alue for any nternally reersble rocess between the two states: the alue of the ntegral deends on the end states only. It can be concluded, therefore, that the ntegral defnes the change n some roerty of the system. Selectng the symbol S to denote ths roerty, ts change s gen by S Q S = δ nt re (.4a) where the subscrt nt re ndcates that the ntegraton s carred out for any nternally reersble rocess lnkng the two states. hs extense roerty s called entroy. Snce entroy s a roerty, the change n entroy of a system n gong from one state to another s the same for all rocesses, both nternally reersble and rreersble, between these two states. In other words, once the change n entroy between two states has been ealuated, ths s the magntude of the entroy change for any rocess of the system between these end states. he defnton of entroy change exressed on a dfferental bass s Q ds = δ nt re (.4b) Equaton.4b ndcates that when a closed system undergong an nternally reersble rocess recees energy by heat transfer, the system exerences an ncrease n entroy. Conersely, when energy s remoed from the system by heat transfer, the entroy of the system decreases. hs can be nterreted to mean that an entroy transfer s assocated wth (or accomanes) heat transfer. he drecton of the entroy transfer s the same as that of the heat transfer. In an adabatc nternally reersble rocess of a closed system the entroy would reman constant. A constant entroy rocess s called an sentroc rocess. On rearrangement, Equaton.4b becomes ( δq) nt = ds hen, for an nternally reersble rocess of a closed system between state and state, re Qnt = m ds re (.5) When such a rocess s reresented by a contnuous cure on a lot of temerature s. secfc entroy, the area under the cure s the magntude of the heat transfer er unt of system mass. Entroy Balance For a cycle consstng of an actual rocess from state to state, durng whch nternal rreersbltes are resent, followed by an nternally reersble rocess from state to state, Equaton.3b takes the form δq + b δq S = nt where the frst ntegral s for the actual rocess and the second ntegral s for the nternally reersble rocess. Snce no rreersbltes are assocated wth the nternally reersble rocess, the term S gen accountng for the effect of rreersbltes durng the cycle can be dentfed wth the actual rocess only. re gen

13 Alyng the defnton of entroy change, the second ntegral of the foregong equaton can be exressed as S Q S = δ Introducng ths and rearrangng the equaton, the closed system entroy balance results: nt re S Q S = S + δ b gen (.6) entroy change entroy transfer entroy generaton When the end states are fxed, the entroy change on the left sde of Equaton.6 can be ealuated ndeendently of the detals of the rocess from state to state. Howeer, the two terms on the rght sde deend exlctly on the nature of the rocess and cannot be determned solely from knowledge of the end states. he frst term on the rght sde s assocated wth heat transfer to or from the system durng the rocess. hs term can be nterreted as the entroy transfer assocated wth (or accomanyng) heat transfer. he drecton of entroy transfer s the same as the drecton of the heat transfer, and the same sgn conenton ales as for heat transfer: a oste alue means that entroy s transferred nto the system, and a negate alue means that entroy s transferred out. he entroy change of a system s not accounted for solely by entroy transfer, but s also due to the second term on the rght sde of Equaton.6 denoted by S gen. he term S gen s oste when nternal rreersbltes are resent durng the rocess and anshes when nternal rreersbltes are absent. hs can be descrbed by sayng that entroy s generated (or roduced) wthn the system by the acton of rreersbltes. he second law of thermodynamcs can be nterreted as secfyng that entroy s generated by rreersbltes and consered only n the lmt as rreersbltes are reduced to zero. Snce S gen measures the effect of rreersbltes resent wthn a system durng a rocess, ts alue deends on the nature of the rocess and not solely on the end states. Entroy generaton s not a roerty. When alyng the entroy balance, the obecte s often to ealuate the entroy generaton term. Howeer, the alue of the entroy generaton for a gen rocess of a system usually does not hae much sgnfcance by tself. he sgnfcance s normally determned through comarson. For examle, the entroy generaton wthn a gen comonent mght be comared to the entroy generaton alues of the other comonents ncluded n an oerall system formed by these comonents. By comarng entroy generaton alues, the comonents where arecable rreersbltes occur can be dentfed and rank ordered. hs allows attenton to be focused on the comonents that contrbute most healy to neffcent oeraton of the oerall system. o ealuate the entroy transfer term of the entroy balance requres nformaton regardng both the heat transfer and the temerature on the boundary where the heat transfer occurs. he entroy transfer term s not always subect to drect ealuaton, howeer, because the requred nformaton s ether unknown or undefned, such as when the system asses through states suffcently far from equlbrum. In ractcal alcatons, t s often conenent, therefore, to enlarge the system to nclude enough of the mmedate surroundngs that the temerature on the boundary of the enlarged system corresonds to the ambent temerature, amb. he entroy transfer term s then smly Q/ amb. Howeer, as the rreersbltes resent would not be ust those for the system of nterest but those for the enlarged system, the entroy generaton term would account for the effects of nternal rreersbltes wthn the

14 system and external rreersbltes resent wthn that orton of the surroundngs ncluded wthn the enlarged system. A form of the entroy balance conenent for artcular analyses s the rate form: ds dt Q = + S gen (.7) where ds/dt s the tme rate of change of entroy of the system. he term Q / reresents the tme rate of entroy transfer through the orton of the boundary whose nstantaneous temerature s. he term Ṡ gen accounts for the tme rate of entroy generaton due to rreersbltes wthn the system. For a system solated from ts surroundngs, the entroy balance s = S S S sol gen (.8) where S gen s the total amount of entroy generated wthn the solated system. Snce entroy s generated n all actual rocesses, the only rocesses of an solated system that actually can occur are those for whch the entroy of the solated system ncreases. hs s known as the ncrease of entroy rncle.

15 . Control Volume Alcatons Snce most alcatons of engneerng thermodynamcs are conducted on a control olume bass, the control olume formulatons of the mass, energy, and entroy balances resented n ths secton are esecally mortant. hese are gen here n the form of oerall balances. Equatons of change for mass, energy, and entroy n the form of dfferental equatons are also aalable n the lterature (see, e.g., Brd et al., 960). Conseraton of Mass When aled to a control olume, the rncle of mass conseraton states: he tme rate of accumulaton of mass wthn the control olume equals the dfference between the total rates of mass flow n and out across the boundary. An mortant case for engneerng ractce s one for whch nward and outward flows occur, each through one or more orts. For ths case the conseraton of mass rncle takes the form dm dt c = m e m e (.9) he left sde of ths equaton reresents the tme rate of change of mass contaned wthn the control olume, ṁ denotes the mass flow rate at an nlet, and ṁ e s the mass flow rate at an outlet. he olumetrc flow rate through a orton of the control surface wth area da s the roduct of the elocty comonent normal to the area, n, tmes the area: n da. he mass flow rate through da s ρ( n da). he mass rate of flow through a ort of area A s then found by ntegraton oer the area ṁ = ρ n A For one-dmensonal flow the ntense roertes are unform wth oston oer area A, and the last equaton becomes da A ṁ = ρa = (.0) where denotes the secfc olume and the subscrt n has been droed from elocty for smlcty. Control Volume Energy Balance When aled to a control olume, the rncle of energy conseraton states: he tme rate of accumulaton of energy wthn the control olume equals the dfference between the total ncomng rate of energy transfer and the total outgong rate of energy transfer. Energy can enter and ext a control olume by work and heat transfer. Energy also enters and exts wth flowng streams of matter. Accordngly, for a control olume wth one-dmensonal flow at a sngle nlet and a sngle outlet, d U + KE + PE dt c e = Q c W + mu + + mue e + + gz gz (.)

16 where the underlned terms account for the secfc energy of the ncomng and outgong streams. he terms Q c and Ẇ account, resectely, for the net rates of energy transfer by heat and work oer the boundary (control surface) of the control olume. Because work s always done on or by a control olume where matter flows across the boundary, the quantty Ẇ of Equaton. can be exressed n terms of two contrbutons: one s the work assocated wth the force of the flud ressure as mass s ntroduced at the nlet and remoed at the ext. he other, denoted as Ẇ c, ncludes all other work effects, such as those assocated wth rotatng shafts, dslacement of the boundary, and electrcal effects. he work rate concet of mechancs allows the frst of these contrbutons to be ealuated n terms of the roduct of the ressure force, A, and elocty at the ont of alcaton of the force. o summarze, the work term Ẇ of Equaton. can be exressed (wth Equaton.0) as ṁ ṁ e W W c e Ae e A = W m m c + e( e e ) ( ) = + (.) he terms ( ) and ( e e ) account for the work assocated wth the ressure at the nlet and outlet, resectely, and are commonly referred to as flow work. Substtutng Equaton. nto Equaton., and ntroducng the secfc enthaly h, the followng form of the control olume energy rate balance results: d U + KE + PE dt c e = Q c W c + m h + + me he e + + gz gz (.3) o allow for alcatons where there may be seeral locatons on the boundary through whch mass enters or exts, the followng exresson s arorate: d U + KE + PE dt c e = Q c W c + m h + + me he e + + gz gz e (.4) Equaton.4 s an accountng rate balance for the energy of the control olume. It states that the tme rate of accumulaton of energy wthn the control olume equals the dfference between the total rates of energy transfer n and out across the boundary. he mechansms of energy transfer are heat and work, as for closed systems, and the energy accomanyng the enterng and extng mass. Control Volume Entroy Balance Lke mass and energy, entroy s an extense roerty. And lke mass and energy, entroy can be transferred nto or out of a control olume by streams of matter. As ths s the rncal dfference between the closed system and control olume forms, the control olume entroy rate balance s obtaned by modfyng Equaton.7 to account for these entroy transfers. he result s ds dt c Q = + ms ms e e + S e gen (.5) rate of entroy change rate of entroy transfer rate of entroy generaton

17 where ds c /dt reresents the tme rate of change of entroy wthn the control olume. he terms and ṁs e e account, resectely, for rates of entroy transfer nto and out of the control olume assocated wth mass flow. One-dmensonal flow s assumed at locatons where mass enters and exts. Q reresents the tme rate of heat transfer at the locaton on the boundary where the nstantaneous temerature s ; and Q / accounts for the assocated rate of entroy transfer. Ṡ gen denotes the tme rate of entroy generaton due to rreersbltes wthn the control olume. When a control olume comrses a number of comonents, s the sum of the rates of entroy generaton of the comonents. Ṡ gen Control Volumes at Steady State Engneerng systems are often dealzed as beng at steady state, meanng that all roertes are unchangng n tme. For a control olume at steady state, the dentty of the matter wthn the control olume changes contnuously, but the total amount of mass remans constant. At steady state, Equaton.9 reduces to ṁs m = e m e (.6a) he energy rate balance of Equaton.4 becomes, at steady state, 0 = Q W m h m e c c gz e he gze e (.6b) At steady state, the entroy rate balance of Equaton.5 reads Q 0 = + ms ms + S e e e gen (.6c) Mass and energy are consered quanttes, but entroy s not generally consered. Equaton.6a ndcates that the total rate of mass flow nto the control olume equals the total rate of mass flow out of the control olume. Smlarly, Equaton.6b states that the total rate of energy transfer nto the control olume equals the total rate of energy transfer out of the control olume. Howeer, Equaton.6c shows that the rate at whch entroy s transferred out exceeds the rate at whch entroy enters, the dfference beng the rate of entroy generaton wthn the control olume owng to rreersbltes. Alcatons frequently nole control olumes hang a sngle nlet and a sngle outlet, as, for examle, the control olume of Fgure. where heat transfer (f any) occurs at b : the temerature, or a sutable aerage temerature, on the boundary where heat transfer occurs. For ths case the mass rate balance, Equaton.6a, reduces to m = me. Denotng the common mass flow rate by m, Equatons.6b and.6c read, resectely, 0 = Q + c Wc m ( h he )+ + gz z e Q c 0 = + ms ( s)+ S b e gen ( e) (.7a) (.8a) When Equatons.7a and.8a are aled to artcular cases of nterest, addtonal smlfcatons are usually made. he heat transfer term s droed when t s nsgnfcant relate to other energy Q c

18 FIGURE. One-nlet, one-outlet control olume at steady state. transfers across the boundary. hs may be the result of one or more of the followng: () the outer surface of the control olume s nsulated, () the outer surface area s too small for there to be effecte heat transfer, (3) the temerature dfference between the control olume and ts surroundngs s small enough that the heat transfer can be gnored, (4) the gas or lqud asses through the control olume so quckly that there s not enough tme for sgnfcant heat transfer to occur. he work term Ẇ c dros out of the energy rate balance when there are no rotatng shafts, dslacements of the boundary, electrcal effects, or other work mechansms assocated wth the control olume beng consdered. he changes n knetc and otental energy of Equaton.7a are frequently neglgble relate to other terms n the equaton. he secal forms of Equatons.7a and.8a lsted n able. are obtaned as follows: when there s no heat transfer, Equaton.8a ges s e S gen s = 0 m ( no heat transfer) (.8b) Accordngly, when rreersbltes are resent wthn the control olume, the secfc entroy ncreases as mass flows from nlet to outlet. In the deal case n whch no nternal rreersbltes are resent, mass asses through the control olume wth no change n ts entroy that s, sentrocally. For no heat transfer, Equaton.7a ges W m c = ( h he )+ + gz e ( ze) (.7b) A secal form that s alcable, at least aroxmately, to comressors, ums, and turbnes results from drong the knetc and otental energy terms of Equaton.7b, leang W = m h h c e ( comressors, ums, and turbnes) (.7c)

19 ABLE. Energy and Entroy Balances for One-Inlet, One- Outlet Control Volumes at Steady State and No Heat ransfer Energy balance W m e c = ( h he )+ + gz z Comressors, ums, and turbnes a hrottlng Nozzles, dffusers b Entroy balance W = m h h c e h e h = + h h e e s e S gen s = 0 m ( e) (.7b) (.7c) (.7d) (.7f) (.8b) a b For an deal gas wth constant c, Equaton of able.7 allows Equaton.7c to be wrtten as W (.7c ) c = mc ( e ) he ower deeloed n an sentroc rocess s obtaned wth Equaton 5 of able.7 as W mc ( k ) k c e ( s = c) (.7c ) where c = kr/(k ). For an deal gas wth constant c, Equaton of able.7 allows Equaton.7f to be wrtten as = = + c e e (.7f ) he ext elocty for an sentroc rocess s obtaned wth Equaton 5 of able.7 as k k e c ( ) e s= c where c = kr/(k ). = + (.7f ) In throttlng deces a sgnfcant reducton n ressure s acheed smly by ntroducng a restrcton nto a lne through whch a gas or lqud flows. For such deces Ẇ c = 0 and Equaton.7c reduces further to read h e h ( throttlng rocess) (.7d) hat s, ustream and downstream of the throttlng dece, the secfc enthales are equal. A nozzle s a flow assage of aryng cross-sectonal area n whch the elocty of a gas or lqud ncreases n the drecton of flow. In a dffuser, the gas or lqud decelerates n the drecton of flow. For such deces, Ẇ c = 0. he heat transfer and otental energy change are also generally neglgble. hen Equaton.7b reduces to 0 = h h + e e (.7e)

20 Solng for the outlet elocty = + h h e e ( nozzle, dffuser) (.7f) Further dscusson of the flow-through nozzles and dffusers s roded n Chater. he mass, energy, and entroy rate balances, Equatons.6, can be aled to control olumes wth multle nlets and/or outlets, as, for examle, cases nolng heat-recoery steam generators, feedwater heaters, and counterflow and crossflow heat exchangers. ransent (or unsteady) analyses can be conducted wth Equatons.9,.4, and.5. Illustratons of all such alcatons are roded by Moran and Sharo (000). Examle A turbne recees steam at 7 MPa, 440 C and exhausts at 0. MPa for subsequent rocess heatng duty. If heat transfer and knetc/otental energy effects are neglgble, determne the steam mass flow rate, n kg/hr, for a turbne ower outut of 30 MW when (a) the steam qualty at the turbne outlet s 95%, (b) the turbne exanson s nternally reersble. Soluton. Wth the ndcated dealzatons, Equaton.7c s arorate. Solng, m = W c /( h he ). Steam table data (able A.5) at the nlet condton are h = 36.7 kj/kg, s = 6.60 kj/kg K. (a) At 0. MPa and x = 0.95, h e = kj/kg. hen m = 3 30 MW 0 kj sec sec (.. ) kj kg MW hr = 6, 357 kg hr (b) For an nternally reersble exanson, Equaton.8b reduces to ge s e = s. For ths case, h e = kj/kg (x = 0.906), and ṁ = 4,74 kg/hr. Examle Ar at 500 F, 50 lbf/n., and 0 ft/sec exands adabatcally through a nozzle and exts at 60 F, 5 lbf/n.. For a mass flow rate of 5 lb/sec determne the ext area, n n.. Reeat for an sentroc exanson to 5 lbf/n.. Model the ar as an deal gas (Secton.3, Ideal Gas Model) wth secfc heat c = 0.4 Btu/lb R (k =.4). Soluton. he nozzle ext area can be ealuated usng Equaton.0, together wth the deal gas equaton, = R/: A e ( e) mν mr e e = = he ext elocty requred by ths exresson s obtaned usng Equaton.7f of able., = + c e e ft = 0 s = ft sec e e Btu ft lbf lb ft sec R lb R ( 440 ) Btu lbf

21 FIGURE. Oen feedwater heater. Fnally, wth R = R /M = ft lbf/lb R, Usng Equaton.7f n able. for the sentroc exanson, = ft sec hen A e = 3.9 n.. Examle 3 lb ft lbf R sec lb R A e = ft 99 5 lbf. 5 sec n = 40. n. e = + Fgure. rodes steady-state oeratng data for an oen feedwater heater. Ignorng heat transfer and knetc/otental energy effects, determne the rato of mass flow rates, m / m. Soluton. For ths case Equatons.6a and.6b reduce to read, resectely, Combnng and solng for the rato m + m = m 3 0 = mh + mh mh m / m, 3 3 m m h = h Insertng steam table data, n kj/kg, from able A.5, m m Internally Reersble Heat ransfer and Work For one-nlet, one-outlet control olumes at steady state, the followng exressons ge the heat transfer rate and ower n the absence of nternal rreersbltes: h h = =

22 Q c m = ds nt re W c d g z z m = ν + + nt re (.9) (.30a) (see, e.g., Moran and Sharo, 000). If there s no sgnfcant change n knetc or otental energy from nlet to outlet, Equaton.30a reads W c d ke e m = ν = = 0 nt re (.30b) he secfc olume remans aroxmately constant n many alcatons wth lquds. hen Equaton.30b becomes W c m = ( ) ( = constant) nt re (.30c) When the states sted by a unt of mass flowng wthout rreersbltes from nlet to outlet are descrbed by a contnuous cure on a lot of temerature s. secfc entroy, Equaton.9 mles that the area under the cure s the magntude of the heat transfer er unt of mass flowng. When such an deal rocess s descrbed by a cure on a lot of ressure s. secfc olume, as shown n Fgure.3, the magntude of the ntegral d of Equatons.30a and.30b s reresented by the area a-b-c-d behnd the cure. he area a-b-c -d under the cure s dentfed wth the magntude of the ntegral d of Equaton.0. FIGURE.3 Internally reersble rocess on coordnates.

23 .3 Proerty Relatons and Data Pressure, temerature, olume, and mass can be found exermentally. he relatonshs between the secfc heats c and c and temerature at relately low ressure are also accessble exermentally, as are certan other roerty data. Secfc nternal energy, enthaly, and entroy are among those roertes that are not so readly obtaned n the laboratory. Values for such roertes are calculated usng exermental data of roertes that are more amenable to measurement, together wth arorate roerty relatons dered usng the rncles of thermodynamcs. In ths secton roerty relatons and data sources are consdered for smle comressble systems, whch nclude a wde range of ndustrally mortant substances. Proerty data are roded n the ublcatons of the Natonal Insttute of Standards and echnology (formerly the U.S. Bureau of Standards), of rofessonal grous such as the Amercan Socety of Mechancal Engneerng (ASME), the Amercan Socety of Heatng. Refrgeratng, and Ar Condtonng Engneers (ASHRAE), and the Amercan Chemcal Socety, and of cororate enttes such as Duont and Dow Chemcal. Handbooks and roerty reference olumes such as ncluded n the lst of references for ths chater are readly accessed sources of data. Proerty data are also retreable from arous commercal onlne data bases. Comuter software s ncreasngly aalable for ths urose as well. Basc Relatons for Pure Substances An energy balance n dfferental form for a closed system undergong an nternally reersble rocess n the absence of oerall system moton and the effect of graty reads From Equaton.4b, δq re = ds. When consderaton s lmted to smle comressble systems: systems for whch the only sgnfcant work n an nternally reersble rocess s assocated wth olume change, = dv, the followng equaton s obtaned: ( δw)nt re nt du δq nt δw = re nt re du = ds dv (.3a) Introducng enthaly, H = U + V, the Helmholtz functon, Ψ = U S, and the Gbbs functon, G = H S, three addtonal exressons are obtaned: dh = ds + Vd dψ= dv Sd dg = Vd Sd (.3b) (.3c) (.3d) Equatons.3 can be exressed on a er-unt-mass bass as du = ds d dh = ds + d dψ= d sd dg = d sd (.3a) (.3b) (.3c) (.3d)

24 Smlar exressons can be wrtten on a er-mole bass. Maxwell Relatons Snce only roertes are noled, each of the four dfferental exressons gen by Equatons.3 s an exact dfferental exhbtng the general form dz = M(x, y)dx + N(x, y)dy, where the second mxed artal derates are equal: ( M/ y) = ( N/ x). Underlyng these exact dfferentals are, resectely, functons of the form u(s, ), h(s, ), ψ(, ), and g(, ). From such consderatons the Maxwell relatons gen n able. can be establshed. Examle 4 Dere the Maxwell relaton followng from Equaton.3a. ABLE. Relatons from Exact Dfferentals

25 Soluton. he dfferental of the functon u = u(s, ) s By comarson wth Equaton.3a, u u du = ds d + s u u = = s, In Equaton.3a, lays the role of M and lays the role of N, so the equalty of second mxed artal derates ges the Maxwell relaton, = s s Snce each of the roertes,,, and s aears on the rght sde of two of the eght coeffcents of able., four addtonal roerty relatons can be obtaned by equatng such exressons: u h = s, s hese four relatons are dentfed n able. by brackets. As any three of Equatons.3 can be obtaned from the fourth smly by manulaton, the 6 roerty relatons of able. also can be regarded as followng from ths sngle dfferental exresson. Seeral addtonal frst-derate roerty relatons can be dered; see, e.g., Zemansky, 97. Secfc Heats and Other Proertes Engneerng thermodynamcs uses a wde assortment of thermodynamc roertes and relatons among these roertes. able.3 lsts seeral commonly encountered roertes. Among the entres of able.3 are the secfc heats c and c. hese ntense roertes are often requred for thermodynamc analyss, and are defned as artal deratons of the functons u(, ) and h(, ), resectely, u ψ = s h g =, ψ g = s s s c c u = h = (.33) (.34) Snce u and h can be exressed ether on a unt mass bass or a er-mole bass, alues of the secfc heats can be smlarly exressed. able.4 summarzes relatons nolng c and c. he roerty k, the secfc heat rato, s c k = c (.35)

26 ABLE.3 Symbols and Defntons for Selected Proertes Proerty Symbol Defnton Proerty Symbol Defnton Pressure Secfc heat, constant olume c emerature Secfc heat, constant ressure c Secfc olume Volume exansty β Secfc nternal energy u Isothermal comressty κ Secfc entroy s Isentroc comressblty α Secfc enthaly h u + Isothermal bulk modulus B Secfc Helmholtz functon ψ u s Isentroc bulk modulus B s Secfc Gbbs functon g h s Joule-homson coeffcent µ J Comressblty factor Z /R Joule coeffcent η Secfc heat rato k c /c Velocty of sound c ( u ) ( h ) ( ) ( ) ( ) s ( ) ( ) ( ) h ( ) s u ( ) s Values for c and c can be obtaned a statstcal mechancs usng sectroscoc measurements. hey can also be determned macroscocally through exactng roerty measurements. Secfc heat data for common gases, lquds, and solds are roded by the handbooks and roerty reference olumes lsted among the Chater references. Secfc heats are also consdered n Secton.3 as a art of the dscussons of the ncomressble model and the deal gas model. Fgure.4 shows how c for water aor ares as a functon of temerature and ressure. Other gases exhbt smlar behaor. he fgure also ges the araton of c wth temerature n the lmt as ressure tends to zero (the deal gas lmt). In ths lmt c ncreases wth ncreasng temerature, whch s a characterstc exhbted by other gases as well he followng two equatons are often conenent for establshng relatons among roertes: x y y = x z y z x z x = y x y z z (.36a) (.36b) her use s llustrated n Examle 5. Examle 5 Obtan Equatons and of able.4 from Equaton. Soluton. Identfyng x, y, z wth s,, and, resectely, Equaton.36b reads s s = s Alyng Equaton.36a to each of ( / ) s and ( / s),

27 FIGURE.4 c of water aor as a functon of temerature and ressure. (Adated from Keenan, J.H., Keyes, F.G., Hll, P.G., and Moore, J.G. 969 and 978. Steam ables S.I. Unts (Englsh Unts). John Wley & Sons, New York.)

28 Introducng the Maxwell relaton from able. corresondng to ψ(, ), Wth ths, Equaton of able.4 s obtaned from Equaton, whch n turn s obtaned n Examle 6. Equaton of able.4 can be obtaned by dfferentatng Equaton wth reect to secfc olume at fxed temerature, and agan usng the Maxwell relaton corresondng to ψ. ABLE.4 Secfc Heat Relatons a () () (3) (4) (5) (6) (7) (8) (9) (0) () () a See, for examle, Moran, M.J. and Sharo, H.N Fundamentals of Engneerng hermodynamcs, 4th ed. Wley, New York. c u s = = = s c h s = = = s c c = = = β κ c J = µ c = η k c c s = = c = c = s s s s s = = s s =

29 P-- Relatons Consderable ressure, secfc olume, and temerature data hae been accumulated for ndustrally mortant gases and lquds. hese data can be reresented n the form = f (, ), called an equaton of state. Equatons of state can be exressed n tabular, grahcal, and analytcal forms. P-- Surface he grah of a functon = f (, ) s a surface n three-dmensonal sace. Fgure.5 shows the -- relatonsh for water. Fgure.5b shows the roecton of the surface onto the ressure-temerature lane, called the hase dagram. he roecton onto the lane s shown n Fgure.5c. FIGURE.5 Pressure-secfc olume-temerature surface and roectons for water (not to scale). Fgure.5 has three regons labeled sold, lqud, and aor where the substance exsts only n a sngle hase. Between the sngle hase regons le two-hase regons, where two hases coexst n equlbrum. he lnes searatng the sngle-hase regons from the two-hase regons are saturaton lnes. Any state reresented by a ont on a saturaton lne s a saturaton state. he lne searatng the lqud hase and

30 the two-hase lqud-aor regon s the saturated lqud lne. he state denoted by f s a saturated lqud state. he saturated aor lne searates the aor regon and the two-hase lqud-aor regon. he state denoted by g s a saturated aor state. he saturated lqud lne and the saturated aor lne meet at the crtcal ont. At the crtcal ont, the ressure s the crtcal ressure c, and the temerature s the crtcal temerature c. hree hases can coexst n equlbrum along the lne labeled trle lne. he trle lne roects onto a ont on the hase dagram. he trle ont of water s used n defnng the Keln temerature scale (Secton., Basc Concets and Defntons; he Second Law of hermodynamcs, Entroy). When a hase change occurs durng constant ressure heatng or coolng, the temerature remans constant as long as both hases are resent. Accordngly, n the two-hase lqud-aor regon, a lne of constant ressure s also a lne of constant temerature. For a secfed ressure, the corresondng temerature s called the saturaton temerature. For a secfed temerature, the corresondng ressure s called the saturaton ressure. he regon to the rght of the saturated aor lne s known as the suerheated aor regon because the aor exsts at a temerature greater than the saturaton temerature for ts ressure. he regon to the left of the saturated lqud lne s known as the comressed lqud regon because the lqud s at a ressure hgher than the saturaton ressure for ts temerature. When a mxture of lqud and aor coexsts n equlbrum, the lqud hase s a saturated lqud and the aor hase s a saturated aor. he total olume of any such mxture s V = V f + V g ; or, alternately, m = m f f + m g g, where m and denote mass and secfc olume, resectely. Ddng by the total mass of the mxture m and lettng the mass fracton of the aor n the mxture, m g /m, be symbolzed by x, called the qualty, the aarent secfc olume of the mxture s + = x x f g = + x f fg (.37a) where fg = g f. Exressons smlar n form can be wrtten for nternal energy, enthaly, and entroy: + u = x u xu f g = u + xu f fg + h = x h xh f g = h + xh f fg + s = x s xs f g = s + xs f fg (.37b) (.37c) (.37d) For the case of water, Fgure.6 llustrates the hase change from sold to lqud (meltng): a-b-c; from sold to aor (sublmaton): a -b -c ; and from lqud to aor (aorzaton): a -b -c. Durng any such hase change the temerature and ressure reman constant and thus are not ndeendent roertes. he Claeyron equaton allows the change n enthaly durng a hase change at fxed temerature to be ealuated from -- data ertanng to the hase change. For aorzaton, the Claeyron equaton reads d d sat hg hf = ( g f) (.38)

31 FIGURE.6 Phase dagram for water (not to scale). where (d/d) sat s the sloe of the saturaton ressure-temerature cure at the ont determned by the temerature held constant durng the hase change. Exressons smlar n form to Equaton.38 can be wrtten for sublmaton and meltng. he Claeyron equaton shows that the sloe of a saturaton lne on a hase dagram deends on the sgns of the secfc olume and enthaly changes accomanyng the hase change. In most cases, when a hase change takes lace wth an ncrease n secfc enthaly, the secfc olume also ncreases, and (d/d) sat s oste. Howeer, n the case of the meltng of ce and a few other substances, the secfc olume decreases on meltng. he sloe of the saturated sold-lqud cure for these few substances s negate, as llustrated for water n Fgure.6. Grahcal Reresentatons he ntense states of a ure, smle comressble system can be reresented grahcally wth any two ndeendent ntense roertes as the coordnates, excludng roertes assocated wth moton and graty. Whle any such ar may be used, there are seeral selectons that are conentonally emloyed. hese nclude the - and - dagrams of Fgure.5, the -s dagram of Fgure.7, the h-s (Moller) dagram of Fgure.8, and the -h dagram of Fgure.9. he comressblty charts consdered next use the comressblty factor as one of the coordnates. Comressblty Charts he -- relaton for a wde range of common gases s llustrated by the generalzed comressblty chart of Fgure.0. In ths chart, the comressblty factor, Z, s lotted s. the reduced ressure, R, reduced temerature, R, and seudoreduced secfc olume, R, where Z = R (.39) and

32 FIGURE.7 emerature-entroy dagram for water. (Source: Jones, J.B. and Dugan, R.E Engneerng hermodynamcs, Prentce-Hall, Englewood Clffs, NJ, based on data and formulatons from Haar, L., Gallagher, J.S., and Kell, G.S NBS/NRC Steam ables. Hemshere, Washngton, D.C.)

33 FIGURE.8 Enthaly-entroy (Moller) dagram for water. (Source: Jones, J.B. and Dugan, R.E Engneerng hermodynamcs. Prentce-Hall, Englewood Clffs, NJ, based on data and formulatons from Haar, L., Gallagher, J.S., and Kell, G.S NBS/NRC Steam ables. Hemshere, Washngton, D.C.) R =, R =, R c = c ( Rc c) (.40) In these exressons, R s the unersal gas constant and c and c denote the crtcal ressure and temerature, resectely. Values of c and c are gen for seeral substances n able A.9. he reduced sotherms of Fgure.0 reresent the best cures ftted to the data of seeral gases. For the 30 gases used n deelong the chart, the deaton of obsered alues from those of the chart s at most on the order of 5% and for most ranges s much less. * Fgure.0 ges a common alue of about 0.7 for the comressblty factor at the crtcal ont. As the crtcal comressblty factor for dfferent substances actually ares from 0.3 to 0.33, the chart s naccurate n the cnty of the crtcal ont. hs source of naccuracy can be remoed by restrctng the correlaton to substances hang essentally the same Z c alues. whch s equalent to ncludng the crtcal comressblty factor as an ndeendent arable: Z = f ( R, R, Z c ). o achee greater accuracy * o determne Z for hydrogen, helum, and neon aboe a R of 5, the reduced temerature and ressure should be calculated usng R = /( c + 8) and P R = /( c + 8), where temeratures are n K and ressures are n atm.

34 FIGURE.9 Pressure-enthaly dagram for water. (Source: Jones, J.B. and Dugan, R.E Engneerng hermodynamcs. Prentce-Hall, Englewood Clffs, NJ, based on data and formulatons from Haar, L., Gallagher, J.S., and Kell, G.S NBS/NRC Steam ables. Hemshere, Washngton, D.C.)

35 FIGURE.0Generalzed comressblty chart ( R = / C, R = / C, R = C /R C ) for R 0. (Source: Obert, E.F. 960 Concets of hermodynamcs. McGraw- Hll, New York.)

36 arables other than Z c hae been roosed as a thrd arameter for examle, the acentrc factor (see, e.g., Red and Sherwood, 966). Generalzed comressblty data are also aalable n tabular form (see, e.g., Red and Sherwood, 966) and n equaton form (see, e.g., Reynolds, 979). he use of generalzed data n any form (grahcal, tabular, or equaton) allows,, and for gases to be ealuated smly and wth reasonable accuracy. When accuracy s an essental consderaton, generalzed comressblty data should not be used as a substtute for -- data for a gen substance as roded by comuter software, a table, or an equaton of state. Equatons of State Consderng the sotherms of Fgure.0, t s lausble that the araton of the comressblty factor mght be exressed as an equaton, at least for certan nterals of and. wo exressons can be wrtten that enoy a theoretcal bass. One ges the comressblty factor as an nfnte seres exanson n ressure, Z = + Bˆ ( ) + Cˆ ( ) + Dˆ 3 ( ) + K and the other s a seres n /, Z B C D K 3 = hese exressons are known as ral exansons, and the coeffcents BCD ˆ, ˆ ˆ, and B, C, D are called ral coeffcents. In rncle, the ral coeffcents can be calculated usng exressons from statstcal mechancs dered from consderaton of the force felds around the molecules. hus far only the frst few coeffcents hae been calculated and only for gases consstng of relately smle molecules. he coeffcents also can be found, n rncle, by fttng -- data n artcular realms of nterest. Only the frst few coeffcents can be found accurately ths way, howeer, and the result s a truncated equaton ald only at certan states. Oer 00 equatons of state hae been deeloed n an attemt to ortray accurately the -- behaor of substances and yet aod the comlextes nherent n a full ral seres. In general, these equatons exhbt lttle n the way of fundamental hyscal sgnfcance and are manly emrcal n character. Most are deeloed for gases, but some descrbe the -- behaor of the lqud hase, at least qualtately. Eery equaton of state s restrcted to artcular states. he realm of alcablty s often ndcated by gng an nteral of ressure, or densty, where the equaton can be exected to reresent the -- behaor fathfully. When t s not stated, the realm of alcablty often may be aroxmated by exressng the equaton n terms of the comressblty factor Z and the reduced roertes, and suermosng the result on a generalzed comressblty chart or comarng wth comressblty data from the lterature. Equatons of state can be classfed by the number of adustable constants they nole. he Redlch- Kwong equaton s consdered by many to be the best of the two-constant equatons of state. It ges ressure as a functon of temerature and secfc olume and thus s exlct n ressure: R a = b + b (.4) hs equaton s rmarly emrcal n nature, wth no rgorous ustfcaton n terms of molecular arguments. Values for the Redlch-Kwong constants for seeral substances are roded n able A.9. Modfed forms of the equaton hae been roosed wth the am of acheng better accuracy.

37 Although the two-constant Redlch-Kwong equaton erforms better than some equatons of state hang seeral adustable constants, two-constant equatons tend to be lmted n accuracy as ressure (or densty) ncreases. Increased accuracy normally requres a greater number of adustable constants. For examle, the Benedct-Webb-Rubn equaton, whch noles eght adustable constants, has been successful n redctng the -- behaor of lght hydrocarbons. he Benedct-Webb-Rubn equaton s also exlct n ressure, R C = + BR A br a aα c γ γ ex (.4) Values of the Benedct-Webb-Rubn constants for arous gases are roded n the lterature (see, e.g., Cooer and Goldfrank, 967). A modfcaton of the Benedct-Webb-Rubn equaton nolng constants s dscussed by Lee and Kessler, 975. Many multconstant equatons can be found n the engneerng lterature, and wth the adent of hgh seed comuters, equatons hang 50 or more constants hae been deeloed for reresentng the -- behaor of dfferent substances. Gas Mxtures Snce an unlmted arety of mxtures can be formed from a gen set of ure comonents by aryng the relate amounts resent, the roertes of mxtures are reorted only n secal cases such as ar. Means are aalable for redctng the roertes of mxtures, howeer. Most technques for redctng mxture roertes are emrcal n character and are not dered from fundamental hyscal rncles. he realm of aldty of any artcular technque can be establshed by comarng redcted roerty alues wth emrcal data. In ths secton, methods for ealuatng the -- relatons for ure comonents are adated to obtan lausble estmates for gas mxtures. he case of deal gas mxtures s dscussed n Secton.3, Ideal Gas Model. In Secton.3, Multcomonent Systems, some general asects of roerty ealuaton for multcomonent systems are resented. he total number of moles of mxture, n, s the sum of the number of moles of the comonents, n : n = n + n + Kn = n = (.43) he relate amounts of the comonents resent can be descrbed n terms of mole fractons. he mole fracton y of comonent s y = n /n. he sum of the mole fractons of all comonents resent s equal to unty. he aarent molecular weght M s the mole fracton aerage of the comonent molecular weghts, such that M = y M = (.44) he relate amounts of the comonents resent also can be descrbed n terms of mass fractons: m /m, where m s the mass of comonent and m s the total mass of mxture. he -- relaton for a gas mxture can be estmated by alyng an equaton of state to the oerall mxture. he constants aearng n the equaton of state are mxture alues determned wth emrcal combnng rules deeloed for the equaton. For examle, mxture alues of the constants a and b for use n the Redlch-Kwong equaton are obtaned usng relatons of the form a = ya b yb, = = = (.45)

38 where a and b are the alues of the constants for comonent. Combnaton rules for obtanng mxture alues for the constants n other equatons of state are also found n the lterature. Another aroach s to regard the mxture as f t were a sngle ure comonent hang crtcal roertes calculated by one of seeral mxture rules. Kay s rule s erhas the smlest of these, requrng only the determnaton of a mole fracton aeraged crtcal temerature c and crtcal ressure c : = y, = y c c, c c, = = (.46) where c, and c, are the crtcal temerature and crtcal ressure of comonent, resectely. Usng c and c, the mxture comressblty factor Z s obtaned as for a sngle ure comonent. he unkown quantty from among the ressure, olume V, temerature, and total number of moles n of the gas mxture can then be obtaned by solng Z = V/nR. Addtonal means for redctng the -- relaton of a mxture are roded by emrcal mxture rules. Seeral are found n the engneerng lterature. Accordng to the addte ressure rule, the ressure of a gas mxture s exressble as a sum of ressures exerted by the nddual comonents: = + + K 3 ] V, (.47a) where the ressures,, etc. are ealuated by consderng the resecte comonents to be at the olume V and temerature of the mxture. he addte ressure rule can be exressed alternately as Z = y Z = V, (.47b) where Z s the comressblty factor of the mxture and the comressblty factors Z are determned assumng that comonent occues the entre olume of the mxture at the temerature. he addte olume rule ostulates that the olume V of a gas mxture s exressble as the sum of olumes occued by the nddual comonents: V = V + V + V K 3 ], (.48a) where the olumes V, V, etc. are ealuated by consderng the resecte comonents to be at the ressure and temerature of the mxture. he addte olume rule can be exressed alternately as Z = y Z =, (.48b) where the comressblty factors Z are determned assumng that comonent exsts at the ressure and temerature of the mxture. Ealuatng h, u, and s Usng arorate secfc heat and -- data, the changes n secfc enthaly, nternal energy, and entroy can be determned between states of sngle-hase regons. able.5 rodes exressons for such roerty changes n terms of artcular choces of the ndeendent arables: temerature and ressure, and temerature and secfc olume.

39 akng Equaton of able.5 as a reresentate case, the change n secfc enthaly between states and can be determned usng the three stes shown n the accomanyng roerty dagram. hs requres knowledge of the araton of c wth temerature at a fxed ressure, and the araton of [ ( / ) ] wth ressure at temeratures and : -a: Snce temerature s constant at, the frst ntegral of Equaton n able.5 anshes, and h h d a [ ] = a-b: Snce ressure s constant at, the second ntegral of Equaton anshes, and hb ha = c(, ) d b-: Snce temerature s constant at, the frst ntegral of Equaton anshes, and [ ] h hb = ( ) d Addng these exressons, the result s h h. he requred ntegrals may be erformed numercally or analytcally. he analytcal aroach s exedted when an equaton of state exlct n secfc olume s known. Smlar consderatons aly to Equatons to 4 of able.5. o ealuate u u wth Equaton 3, for examle, requres the araton of c wth temerature at a fxed secfc olume, and the araton of [( / ) ] wth secfc olume at temeratures and. An analytcal aroach to erformng the ntegrals s exedted when an equaton of state exlct n ressure s known. As changes n secfc enthaly and nternal energy are related through h = u + by h h = ( u u )+ (.49) only one of h h and u u need be found by ntegraton. he other can be ealuated from Equaton.49. he one found by ntegraton deends on the nformaton aalable: h h would be found when an equaton of state exlct n and c as a functon of temerature at some fxed ressure s known, u u would be found when an equaton of state exlct n and c as a functon of temerature at some secfc olume s known. Examle 6 Obtan Equaton of able.4 and Equatons 3 and 4 of able.5. Soluton. Wth Equaton.33 and the Maxwell relaton corresondng to ψ(, ) from able., Equatons 3 and 4 of able.5 become, resectely, u du = cd + Introducng these exressons for du and ds n Equaton.3a, and collectng terms, d s ds = d d +

40 ABLE.5 h, u, s Exressons

41 s Snce and are ndeendent, the coeffcents of d and d must ansh, gng, resectely, he frst of these corresonds to Equaton of able.4 and Equaton 4 of able.5. he second of the aboe exressons establshes Equaton 3 of able.5. Wth smlar consderatons, Equaton 3 of able.4 and Equatons and of able.5 may be obtaned. Fundamental hermodynamc Functons A fundamental thermodynamc functon s one that rodes a comlete descrton of the thermodynamc state. he functons u(s, ), h(s, ), ψ(, ), and g(, ) lsted n able. are fundamental thermodynamc functons. In rncle, all roertes of nterest can be determned from a fundamental thermodynamc functon by dfferentaton and combnaton. akng the functon ψ(, ) as a reresentate case, the roertes and, beng the ndeendent arables, are secfed to fx the state. he ressure and secfc entroy s at ths state can be determned by dfferentaton of ψ(, ), as shown n able.. By defnton, ψ = u s, so secfc nternal energy s obtaned as wth u,, and known, the secfc enthaly can be found from the defnton h = u +. Smlarly, the secfc Gbbs functon s found from the defnton g = h s. he secfc heat c can be determned by further dfferentaton c = ( u/ ). he deeloment of a fundamental functon requres the selecton of a functonal form n terms of the arorate ar of ndeendent roertes and a set of adustable coeffcents that may number 50 or more. he functonal form s secfed on the bass of both theoretcal and ractcal consderatons. he coeffcents of the fundamental functon are determned by requrng that a set of selected roerty alues and/or obsered condtons be statsfed n a least-squares sense. hs generally noles roerty data requrng the assumed functonal form to be dfferentated one or more tmes, for examle -- and secfc heat data. When all coeffcents hae been ealuated, the functon s tested for accuracy by usng t to ealuate roertes for whch acceted alues are known such as elocty of sound and Joule- homson data. Once a sutable fundamental functon s establshed, extreme accuracy n and consstency among the thermodynamc roertes are ossble. he roertes of water tabulated by Keenan et al. (969) and by Haar et al. (984) hae been calculated from reresentatons of the Helmholtz functon. hermodynamc Data Retreal u c d = + s c = u = u = ψ + s abular resentatons of ressure, secfc olume, and temerature are aalable for ractcally mortant gases and lquds. he tables normally nclude other roertes useful for thermodynamc analyses, such as nternal energy, enthaly, and entroy. he arous steam tables ncluded n the references of ths chater rode examles. Comuter software for retreng the roertes of a wde range of substances s also aalable, as, for examle, the ASME Steam ables (993) and Bornakke and Sonntag (996). d

42 Increasngly, textbooks come wth comuter dsks rodng thermodynamc roerty data for water, certan refrgerants, and seeral gases modeled as deal gases see, e.g., Moran and Sharo (996). he samle steam table data resented n able.6 are reresentate of data aalable for substances commonly encountered n mechancal engneerng ractce. able A.5 and Fgures.7 to.9 rode steam table data for a greater range of states. he form of the tables and fgures, and how they are used are assumed to be famlar. In artcular, the use of lnear nterolaton wth such tables s assumed known. Secfc nternal energy, enthaly, and entroy data are determned relate to arbtrary datums and such datums ary from substance to substance. Referrng to able.6a, the datum state for the secfc nternal energy and secfc entroy of water s seen to corresond to saturated lqud water at 0.0 C (3.0 F), the trle ont temerature. he alue of each of these roertes s set to zero at ths state. If calculatons are erformed nolng only dfferences n a artcular secfc roerty, the datum cancels. When there are changes n chemcal comoston durng the rocess, secal care should be exercsed. he aroach followed when comoston changes due to chemcal reacton s consdered n Secton.4. Lqud water data (see able.6d) suggests that at fxed temerature the araton of secfc olume, nternal energy, and entroy wth ressure s slght. he araton of secfc enthaly wth ressure at fxed temerature s somewhat greater because ressure s exlct n the defnton of enthaly. hs behaor for, u, s, and h s exhbted generally by lqud data and rodes the bass for the followng set of equatons for estmatng roerty data at lqud states from saturated lqud data: (, ) ( ) u (, ) u( ) h, h f f f f[ sat ] + s (, ) s( ) f (.50a) (.50b) (.50c) (.50d) As before, the subscrt f denotes the saturated lqud state at the temerature, and sat s the corresondng saturaton ressure. he underlned term of Equaton.50c s often neglgble, gng h(, ) h f (), whch s used n Examle 3 to ealuate h. In the absence of saturated lqud data, or as an alternate to such data, the ncomressble model can be emloyed: = constant Incomressble model: u = u( ) (.5) hs model s also alcable to solds. Snce nternal energy ares only wth temerature, the secfc heat c s also a functon of only temerature: c () = du/d. Although secfc olume s constant, enthaly ares wth both temerature and ressure, such that h (, )= u + (.5) Dfferentaton of Equaton.5 wth resect to temerature at fxed ressure ges c =c. he common secfc heat s often shown smly as c. Secfc heat and densty data for seeral lquds and solds are

43 ABLE.6 Samle Steam able Data em ( C) Pressure (bar) (a) Proertes of Saturated Water (Lqud-Vaor): emerature able Secfc Volume (m 3 /kg) Internal Energy (kj/kg) Enthaly (kj/kg) Entroy (kj/kg K) Saturated Lqud ( f 0 3 ) Saturated Vaor ( g ) Saturated Lqud (u f ) Saturated Vaor (u g ) Saturated Lqud (h f ) Ea. (h fg ) Saturated Vaor (h g ) Saturated Lqud (s f ) Saturated Vaor (s g ) Pressure (bar) em ( C) (b) Proertes of Saturated Water (Lqud-Vaor): Pressure able Secfc Volume (m 3 /kg) Internal Energy (kj/kg) Enthaly (kj/kg) Entroy (kj/kg K) Saturated Lqud ( f 0 3 ) Saturated Vaor ( g ) Saturated Lqud (u f ) Saturated Vaor (u g ) Saturated Lqud (h f ) Ea. (h fg ) Saturated Vaor (h g ) Saturated Lqud (s f ) Saturated Vaor (s g )

44 ABLE.6 Samle Steam able Data (contnued) (c) Proertes of Suerheated Water Vaor ( C) (m 3 /kg) u(kj/kg) h(kj/kg) s(kj/kg K) (m 3 /kg) u(kj/kg) h(kj/kg) s(kj/kg K) = 0.06 bar = MPa ( sat 36.6 C) = 0.35 bar = MPa ( sat = 7.69 C) Sat (d) Proertes of Comressed Lqud Water ( C) 0 3 (m 3 /kg) u(kj/kg) h(kj/kg) s(kj/kg K) 0 3 (m 3 /kg) u(kj/kg) h(kj/kg) s(kj/kg K) = 5 bar =.5 MPa ( sat 3.99 C) = 50 bar = 5.0 MPa ( sat = C) Sat Source: Moran, M.J. and Sharo, H.N Fundamentals of Engneerng hermodynamcs, 4th ed. Wley, New York, as extracted from Keenan, J. H., Keyes, F.G., Hll, P.G., and Moore, J.G Steam ables. Wley, New York.

45 roded n ables B., C., and C.. As the araton of c wth temerature s slght, c s frequently taken as constant. When the ncomressble model s aled. Equaton.49 takes the form h h = c( ) d + = c ( )+ ae (.53) Also, as Equaton.3a reduces to du = ds, and du = c()d, the change n secfc entroy s s = = c ae c d ln (.54) Ideal Gas Model Insecton of the generalzed comressblty chart, Fgure.0, shows that when R s small, and for many states when R s large, the alue of the comressblty factor Z s close to. In other words, for ressures that are low relate to c, and for many states wth temeratures hgh relate to c, the comressblty factor aroaches a alue of. Wthn the ndcated lmts, t may be assumed wth reasonable accuracy that Z = that s, = R or = R (.55a) where R = R/M s the secfc gas constant. Other forms of ths exresson n common use are V = nr, V = mr (.55b) Referrng to Equaton 3 of able.5, t can be concluded that ( u/ ) anshes dentcally for a gas whose equaton of state s exactly gen by Equaton.55, and thus the secfc nternal energy deends only on temerature. hs concluson s suorted by exermental obseratons begnnng wth the work of Joule, who showed that the nternal energy of ar at low densty deends rmarly on temerature. hese consderatons allow for an deal gas model of each real gas: () the equaton of state s gen by Equaton.55 and () the nternal energy and enthaly are functons of temerature alone. he real gas aroaches the model n the lmt of low reduced ressure. At other states the actual behaor may deart substantally from the redctons of the model. Accordngly, cauton should be exercsed when nokng the deal gas model lest sgnfcant error s ntroduced. Secfc heat data for gases can be obtaned by drect measurement. When extraolated to zero ressure, deal gas-secfc heats result. Ideal gas-secfc heats also can be calculated usng molecular models of matter together wth data from sectroscoc measurements. able A.9 rodes deal gassecfc heat data for a number of substances. he followng deal gas-secfc heat relatons are frequently useful: c c c R = + kr R =, c = k k (.56a) (.56b)

46 where k = c /c. Wth the deal gas model, Equatons to 4 of able.5 ge Equatons to 4 of able.7, resectely. Equaton of able.7 can be exressed alternately usng s () defned by s ( ) 0 c( ) d (.57) as = s s s s R,, ln (.58) Exressons smlar n form to Equatons.56 to.68 can be wrtten on a molar bass. ABLE.7 Ideal Gas Exressons for h, u, and s Varable Secfc Heats Constant Secfc Heats h ( h c d () ( ) ) = h h = c ( ) c s ( s d R () ( ), ) (, )= ln s s c (, ) (, )= ln Rln u ( u c d (3) (3 ) ) = u u = c ( ) c s s d R ( (4) (4 ), ) (, )= + ln s s c R (, ) (, )= ln + ln s = s s = s k k r (5) (5 ) r = = ( ) k r ( ) (6) (6 ) r = = For rocesses of an deal gas between states hang the same secfc entroy, s = s, Equaton.58 ges or wth r = ex[s ()/R] ex[ s ( ) R] = ex s R [ ] = r ( ) r ( s = s ) (.59a) A relaton between the secfc olume and temeratures for two states of an deal gas hang the same secfc entroy can also be deeloed: r = r ( s = s ) (.59b)

47 Equatons.59 are lsted n able.7 as Equatons 5 and 6, resectely. able A.8 rodes a tabular dslay of h, u, s, r, and r s. temerature for ar as an deal gas. abulatons of h, u, and s for seeral other common gases are roded n able A.. Proerty retreal software also rodes such data; see, e.g., Moran and Sharo (000). he use of data from able A.8 for the nozzle of Examle s llustrated n Examle 7. When the deal gas-secfc heats are assumed constant, Equatons to 6 of able.7 become Equatons tο 6, resectely. he secfc heat c s taken as constant n Examle. Examle 7 Usng data from able A.8, ealuate the ext elocty for the nozzle of Examle and comare wth the ext elocty for an sentroc exanson to 5 lbf/n.. Soluton. he ext elocty s gen by Equaton.7f = + h h e e At 960 and 50 R, able A.8 ges, resectely, h = 3.06 Btu/lb and h e = 4.7 Btu/lb. hen e ft = 0 s = 3. 5 ft sec Btu lb ft lbf 3.74 lb ft sec Btu lbf Usng Equaton.59a and r data from able A.8, the secfc enthaly at the ext for an sentroc exanson s found as follows: e = = r e r =. 06 Interolatng wth r data, h e = 9.54 Btu/lb. Wth ths, the ext elocty s 363. ft/sec. he actual ext elocty s about % less than the elocty for an sentroc exanson, the maxmum theoretcal alue. In ths artcular alcaton, there s good agreement n each case between eloctes calculated usng able A.8 data and, as n Examle, assumng c constant. Such agreement cannot be exected generally, howeer. See, for examle, the Brayton cycle data of able.5. Polytroc Processes An nternally reersble rocess descrbed by the exresson n = constant s called a olytroc rocess and n s the olytroc exonent. Although ths exresson can be aled wth real gas data, t most generally aears n ractce together wth the use of the deal gas model. able.8 rodes seeral exressons alcable to olytroc rocesses and the secal forms they take when the deal gas model s assumed. he exressons for d and d hae alcaton to work ealuatons wth Equatons.0 and.30, resectely. In some alcatons t may be arorate to determne n by fttng ressuresecfc olume data. Examle 8 llustrates both the olytroc rocess and the reducton n the comressor work acheable by coolng a gas as t s comressed. Examle 8 A comressor oerates at steady state wth ar enterng at bar, 0 C and extng at 5 bar. (a) If the ar undergoes a olytroc rocess wth n =.3, determne the work and heat transfer, each n kj/kg of ar flowng. Reeat for (b) an sothermal comresson and (c) an sentroc comresson.

48 ABLE.8 Polytroc Processes: n = Constant a General = n = 0: constant ressure n = ± : constant secfc olume n Ideal Gas b n n n () = ( ) = ( ) n = 0: constant ressure n = ± : constant secfc olume n = : constant temerature n = k: constant secfc entroy when k s constant n = n = d = ln () d = R ln ( ) = (3) d ln = d R ln (3 ) nr d = n (5) (5 ) = ( n ) n nr n a For olytroc rocesses of closed systems where olume change s the only work mode, Equatons, 4, and, 4 are alcable wth Equaton.0 to ealuate the work. When each unt of mass assng through a one-nlet, oneext control olume at steady state undergoes a olytroc rocess, Equatons 3, 5, and 3, 5 are alcable wth b n n R ( ) d = d = n n (4) (4 ) n n = n ( ) n n R = n ( ) n d = n ( ) = ( n ) n n n Equatons.30a and.30b to ealuate the ower. Also note that generally, d = n d.

49 Soluton. Usng Equaton 5 of able.8 together wth Equaton.30b, W m c nr = n ( n ) n.. = kj kg K [ ] ( 93 K) () 5 = kj kg (he area behnd rocess - of Fgure., area --a-b, reresents the magntude of the work requred, er unt mass of ar flowng.) Also, Equaton of able.8 ges = 45 K. FIGURE. Internally reersble comresson rocesses. An energy rate balance at steady state and enthaly data from able A.8 ges Q c m W c = + h h m = ( )= kj kg (b) Usng Equaton 3 of able.8 together wth Equaton.30b,

50 W m c = R ln. = 8 34 ln ( 93) 5 = kj kg Area - -a-b on Fgure. reresents the magntude of the work requred, er unt of mass of ar flowng. An energy balance reduces to ge Q / / c m = Wc m = 35.3 kj/kg. (c) For an sentroc comresson, Q c = 0 and an energy rate balance reduces to ge W / c m = (h s h ), where s denotes the ext state. Wth Equaton.59a and r data, h s = kj/kg ( s = 463K). hen W / c m = ( ) = 7.6 kj/kg. Area -s-a-b on Fgure. reresents the magntude of the work requred, er unt of mass of ar flowng. Ideal Gas Mxtures When aled to an deal gas mxture, the addte ressure rule (Secton.3, -- Relatons) s known as the Dalton model. Accordng to ths model, each gas n the mxture acts as f t exsts searately at the olume and temerature of the mxture. Alyng the deal gas equaton of state to the mxture as a whole and to each comonent, V = nr, V = nr, where, the artal ressure of comonent, s the ressure that comonent would exert f n moles occued the full olume V at the temerature. Formng a rato, the artal ressure of comonent s n = n = y (.60) where y s the mole fracton of comonent. he sum of the artal ressures equals the mxture ressure. he nternal energy, enthaly, and entroy of the mxture can be determned as the sum of the resecte roertes of the comonent gases, roded that the contrbuton from each gas s ealuated at the condton at whch the gas exsts n the mxture. On a molar bass, U = nu or u = yu = = H = nh or h = yh = = S = ns or s= ys = = (.6a) (.6b) (.6c) c c he secfc heats and for an deal gas mxture n terms of the corresondng secfc heats of the comonents are exressed smlarly: c = = y c (.6d)

51 c = = y c (.6e) When workng on a mass bass, exressons smlar n form to Equatons.6 can be wrtten usng mass and mass fractons n lace of moles and mole fractons, resectely, and usng u, h, s, c, and c n lace of u, h, s, c, and c, resectely. he nternal energy and enthaly of an deal gas deend only on temerature, and thus the u and h terms aearng n Equatons.6 are ealuated at the temerature of the mxture. Snce entroy deends on two ndeendent roertes, the s terms are ealuated ether at the temerature and the artal ressure of comonent, or at the temerature and olume of the mxture. In the former case S = ns (, ) = = ns (, x ) = (.6) Insertng the exressons for H and S gen by Equatons.6b and.6c nto the Gbbs functon, G = H S, G = nh ( ) ns (, ) = = = ng (, ) = (.63) where the molar-secfc Gbbs functon of comonent s g (, ) = h () s (, ). he Gbbs functon of can be exressed alternately as g(, )= g(, )+ Rln = g (, )+ Rln x ( ) (.64) were s some secfed ressure. Equaton.64 s obtaned by ntegratng Equaton.3d at fxed temerature from ressure to. Most Ar An deal gas mxture of artcular nterest for many ractcal alcatons s most ar. Most ar refers to a mxture of dry ar and water aor n whch the dry ar s treated as f t were a ure comonent. Ideal gas mxture rncles usually aly to most ar. In artcular, the Dalton model s alcable, and so the mxture ressure s the sum of the artal ressures a and of the dry ar and water aor, resectely. Saturated ar s a mxture of dry ar and saturated water aor. For saturated ar, the artal ressure of the water aor equals sat (), whch s the saturaton ressure of water corresondng to the dry-bulb (mxture) temerature. he makeu of most ar can be descrbed n terms of the humdty rato (secfc humdty) and the relate humdty. he bulb of a wet-bulb thermometer s coered wth a wck saturated wth lqud water, and the wet-bulb temerature of an ar-water aor mxture s the temerature ndcated by such a thermometer exosed to the mxture.

52 When a samle of most ar s cooled at constant ressure, the temerature at whch the samle becomes saturated s called the dew ont temerature. Coolng below the dew ont temerature results n the condensaton of some of the water aor ntally resent. When cooled to a fnal equlbrum state at a temerature below the dew ont temerature, the orgnal samle would consst of a gas hase of dry ar and saturated water aor n equlbrum wth a lqud water hase. Psychrometrc charts are lotted wth arous most ar arameters, ncludng the dry-bulb and wetbulb temeratures, the humdty rato, and the relate humdty, usually for a secfed alue of the mxture ressure such as atm. Generalzed Charts for Enthaly, Entroy, and Fugacty he changes n enthaly and entroy between two states can be determned n rncle by correctng the resecte roerty change determned usng the deal gas model. he correctons can be obtaned, at least aroxmately, by nsecton of the generalzed enthaly correcton and entroy correcton charts, Fgures. and.3, resectely. Such data are also aalable n tabular form (see, e.g., Red and Sherwood, 966) and calculable usng a generalzed equaton for the comressblty factor (Reynolds, 979). Usng the suerscrt * to dentfy deal gas roerty alues, the changes n secfc enthaly and secfc entroy between states and are h h * * h h = h h Rc R c * * h h R c (.65a) * * s s s s R s s = R s * * s R (.65b) he frst underlned term on the rght sde of each exresson reresents the resecte roerty change assumng deal gas behaor. he second underlned term s the correcton that must be aled to the * * deal gas alue to obtan the actual alue. he quanttes ( h h)/ R c and ( s s)/ R at state would be read from the resecte correcton chart or table or calculated, usng the reduced temerature R and reduced ressure R corresondng to the temerature and ressure at state, resectely. * * Smlarly, ( h h)/ R c and ( s s)/ R at state would be obtaned usng R and R. Mxture alues for c and c determned by alyng Kay s rule or some other mxture rule also can be used to enter the generalzed enthaly correcton and entroy correcton charts. Fgure.4 ges the fugacty coeffcent, f/, as a functon of reduced ressure and reduced temerature. he fugacty f lays a smlar role n determnng the secfc Gbbs functon for a real gas as ressure lays for the deal gas. o deelo ths, consder the araton of the secfc Gbbs functon wth ressure at fxed temerature (from able.) g = For an deal gas, ntegraton at fxed temerature ges * g Rln C = + where C() s a functon of ntegraton. o ealuate g for a real gas, fugacty relaces ressure,

53 FIGURE. Generalzed enthaly correcton chart. (Source: Adated from Van Wylen, G. J. and Sonntag, R. E Fundamentals of Classcal hermodynamcs, 3rd ed., Englsh/SI. Wley, New York.) g = Rln f + C( ) In terms of the fugacty coeffcent the dearture of the real gas alue from the deal gas alue at fxed temerature s then

54 FIGURE.3 Generalzed entroy correcton chart. (Source: Adated from Van Wylen, G. J. and Sonntag, R. E Fundamentals of Classcal hermodynamcs, 3rd ed., Englsh/SI. Wley, New York.) * f g g = Rln (.66) As ressure s reduced at fxed temerature, f/ tends to unty, and the secfc Gbbs functon s gen by the deal gas alue.

55 FIGURE.4 Generalzed fugacty coeffcent chart. (Source: Van Wylen, G. J. and Sonntag, R. E Fundamentals of Classcal hermodynamcs, 3rd ed., Englsh/SI. Wley, New York.) Multcomonent Systems In ths secton are resented some general asects of the roertes of multcomonent systems consstng of nonreactng mxtures. For a sngle hase multcomonent system consstng of comonents, an extense roerty X may be regarded as a functon of temerature, ressure, and the number of moles of each comonent resent n the mxture: X = X(,, n, n, n ). Snce X s mathematcally homogeneous of degree one n the n s, the functon s exressble as