Vtusolution.in BASIC THERMODYNAMICS. Subject Code: 10ME33 IA Marks: 25 Hours/Week: 04 Exam Hours: 03 Total Hours: 52 Exam Marks: 100

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1 BASIC EMODYNAMICS 0ME BASIC EMODYNAMICS Subject Code: 0ME IA Marks: 5 ours/week: 04 Exam ours: 0 otal ours: 5 Exam Marks: 00 PA-A UNI - Fudametal Cocets & Defiitios: hermodyamics defiitio ad scoe, Microscoic ad Macroscoic aroaches. Some ractical alicatios of egieerig thermodyamic Systems, Characteristics of system boudary ad cotrol surface, examles. hermodyamic roerties; defiitio ad uits, itesie ad extesie roerties. hermodyamic state, state oit, state diagram, ath ad rocess, quasi-static rocess, cyclic ad o-cyclic ;rocesses; hermodyamic equilibrium; defiitio, mechaical equilibrium; diathermic wall, thermal equilibrium, chemical equilibrium, Zeroth law of thermodyamics, emerature; cocets, scales, fixed oits ad measuremets. 06 ours UNI - Work ad eat: Mechaics, defiitio of work ad its limitatios. hermodyamic defiitio of work; examles, sig coetio. Dislacemet work; as a art of a system boudary, as a whole of a system boudary, exressios for dislacemet work i arious rocesses through - diagrams. Shaft work; Electrical work. Other tyes of work. eat; defiitio, uits ad sig coetio. 06 ours UNI - First Law of hermodyamics: Joules exerimets, equialece of heat ad work. Statemet of the First law of thermodyamics, extesio of the First law to o - cyclic rocesses, eergy, eergy as a roerty, modes of eergy, ure substace; defiitio, tworoerty rule, Secific heat at costat olume, ethaly, secific heat at costat ressure. Extesio of the First law to cotrol olume; steady state-steady flow eergy equatio, imortat alicatios, aalysis of usteady rocesses such as film ad eacuatio of essels with ad without heat trasfer. 07 ours UNI - 4 Secod Law of hermodyamics: Deices coertig heat to work; (a) i a thermodyamic cycle, (b) i a mechaical cycle. hermal reseroir. Direct heat egie; schematic reresetatio ad efficiecy. Deices coertig work to heat i a thermodyamic cycle; reersed heat egie, schematic reresetatio, coefficiets of erformace. Keli - Plack statemet of the Secod law of hermodyamics; PMM I ad PMM II, Clausius statemet of Secod law of hermodyamics, Equialece of the two statemets; eersible ad irreersible rocesses; factors that make a rocess irreersible, reersible heat egies, Carot cycle, Carot riciles. 07 ours Deartmet of Mechaical Egieerig, Page

2 BASIC EMODYNAMICS 0ME PA-B UNI - 5 Etroy: Clasius iequality; Statemet, roof, alicatio to a reersible cycle. Etroy; defiitio, a roerty, chage of etroy, ricile of icrease i etroy, etroy as a quatitatie test for irreersibility, calculatio of etroy usig ds relatios, etroy as a coordiate. Aailable ad uaailable eergy. 06 ours UNI - 6 Pure Substaces: P- ad P- diagrams, trile oit ad critical oits. Subcooled liquid, saturated liquid, mixture of saturated liquid ad aour, saturated aour ad suerheated aour states of ure substace with water as examle. Ethaly of chage of hase (Latet heat). Dryess fractio (quality), -S ad -S diagrams, reresetatio of arious rocesses o these diagrams. Steam tables ad its use. hrottlig calorimeter, searatig ad throttlig calorimeter. 07 ours UNI - 7 hermodyamic relatios: Maxwell relatio, Clausius Clayero's equatio. Ideal gas; equatio of state, iteral eergy ad ethaly as fuctios of temerature oly, uiersal ad articular gas costats, secific heats, erfect ad semi-erfect gases. Ealuatio of heat, work, chage i iteral eergy. ethaly ad etroy i arious quasi-static rocesses. 07 ours UNI - 8 Ideal gas mixture : Ideal gas mixture; Dalto's laws of artial ressures, Amagat's law of additie olumes, ealuatio of roerties, Aalysis of arious rocesses. eal Gases: Itroductio. a-der Waal's Equatio of state, a-der Waal's costats i terms of critical roerties, Law of corresodig states, comressiblity factor; comressibility chart 06 ours Data adbooks :. hermodyamic data had book, B.. Nijagua.. Proerties of efrigerat & Psychometric (tables & Charts i SI Uits), Dr. S.S. Bawait, Dr. S.C. Laroiya, Birla Pub. Pt. Ltd., Delhi, 008 EX BOOKS:. Basic Egieerig hermodyamics, A.ekatesh, Uiersities Press, 008. Basic ad Alied hermodyamics, P.K.Nag, d Ed., ata McGraw ill Pub. EFEENCE BOOKS:. hermodyamics, A Egieerig Aroach, Yuus A.Ceegal ad Michael A.Boles, ata McGraw ill ublicatios, 00. Egieerig hermodyamics, J.B.Joes ad G.A.awkis, Joh Wiley ad Sos... Fudametals of Classical hermodyamics, G.J.a Wyle ad.e.sotag, Wiley Easter. 4. A Itroductio to hermodyamcis, Y..C.ao, Wiley Easter, 99, 5. B.K ekaa, Swati B. Wadaadagi Basic hermodyamics, PI, New Delhi, 00 Deartmet of Mechaical Egieerig, Page

3 BASIC EMODYNAMICS CONENS 0ME. Fudametal Cocets & Defiitios 4-. Work ad eat -5. First Law of hermodyamics Secod Law of hermodyamics Etroy Pure Substaces hermodyamic relatios Ideal gas mixture 05-9 Deartmet of Mechaical Egieerig, Page

4 BASIC EMODYNAMICS 0ME UNI Itroductio hermodyamics ioles the storage, trasformatio, ad trasfer of eergy. Eergy is stored as iteral eergy (due to temerature), kietic eergy (due to motio), otetial eergy (due to eleatio), ad chemical eergy (due to chemical comositio); it is trasformed from oe of these forms to aother; ad it is trasferred across a boudary as either heat or work. We will reset equatios that relate the trasformatios ad trasfers of eergy to roerties such as temerature, ressure, ad desity. he roerties of materials thus become ery imortat. May equatios will be based o exerimetal obseratios that hae bee reseted as mathematical statemets, or laws: rimarily the first ad secod laws of thermodyamics. he mechaical egieer s objectie i studyig thermodyamics is most ofte the aalysis of a rather comlicated deice, such as a air coditioer, a egie, or a ower lat. As the fluid flows through such a deice, it is assumed to be a cotiuum i which there are measurable quatities such as ressure, temerature, ad elocity. his book, the, will be restricted to macroscoic or egieerig thermodyamics. If the behaior of idiidual molecules is imortat, statistical thermodyamics must be cosulted. System: We eed to fix our focus of attetio i order to uderstad heat ad work iteractio. he body or assemblage or the sace o which our attetio is focused is called system. he system may be haig real or imagiary boudaries across which the iteractio occurs. he boudary may be rigid ad sometimes take differet shaes at differet times. If the system has imagiary boudary the we must roerly formulate the idea of system i our mid. Deartmet of Mechaical Egieerig, Page 4

5 BASIC EMODYNAMICS 0ME Surroudigs: Eerythig else aart from system costitutes surroudigs. he idea of surroudigs gets formulated the momet we defie system. System ad surroudigs together form what is kow as uierse. Closed system: If the system has a boudary through which mass or material caot be trasferred, but oly eergy ca be trasferred is called closed system. I a actual system, there may ot be eergy trasfer. What is essetial for the system to be closed is the iability of the boudary to trasfer mass oly. Oe system: If the system has a boudary through which both eergy ad mass ca trasfer, the it is called oe system. Isolated System A isolated system is that system which exchages either eergy or matter with ay other system or with eiromet. Deartmet of Mechaical Egieerig, Page 5

6 BASIC EMODYNAMICS 0ME Proerties: ariables such as ressure, temerature, olume ad mass are roerties. A system will hae a sigle set of all these alues. Itesie roerties: he roerties that are ideedet of amout cotaied i the system are called extesie roerties. For examle, take temerature. We ca hae a substace with aryig amout but still same temerature. Desity is aother examle of itesie roerty because desity of water is same o matter how much is the water. Other itesie roerties are ressure, iscosity, surface tesio. Extesie roerties: he roerties that deed uo amout cotaied i the system are called extesie roerties. Mass deeds uo how much substace a system has i it therefore mass is a extesie roerty. State: It is defied as coditio of a system i which there are oe set of alues for all its roerties. he roerties that defie the state of a system are called state ariables. here is certai miimum umber of itesie roerties that requires to be secified i order to defie the state of a system ad this umber is uiquely related to the kid of system. his relatio is hase rule which we shall discuss little later. Process: he chages that occur i the system i moig the system from oe state to the other is called a rocess. Durig a rocess the alues of some or all state ariables chage. he rocess may be accomaied by heat or work iteractio with the system. Equilibrium state: A system is said to be i thermodyamic equilibrium if it satisfies the coditio for thermal equilibrium, mechaical equilibrium ad also chemical equilibrium. If it is i equilibrium, there are o chages occurrig or there is o rocess takig lace. hermal equilibrium: here should ot be ay temerature differece betwee differet regios or locatios withi the system. If there are, the there is o way a rocess of heat trasfer does ot take lace. Uiformity of temerature throughout the system is the requiremet for a system to be i thermal equilibrium. Surroudigs ad the system may be at differet temeratures ad still system may be i thermal equilibrium. Mechaical equilibrium: here should ot be ay ressure differece betwee differet regios or locatios withi the system. If there are, the there is o way a rocess of work trasfer does ot take lace. Uiformity of ressure throughout the system is the requiremet for a system to be i mechaical equilibrium. Surroudigs ad the system may be at ressures ad still system may be i mechaical equilibrium. Deartmet of Mechaical Egieerig, Page 6

7 BASIC EMODYNAMICS 0ME Chemical equilibrium: here should ot be ay chemical reactio takig lace aywhere i the system, the it is said to be i chemical equilibrium. Uiformity of chemical otetial throughout the system is the requiremet for a system to be i chemical equilibrium. Surroudigs ad the system may hae differet chemical otetial ad still system may be i chemical equilibrium. hermodyamic rocess: A system i thermodyamic equilibrium is disturbed by imosig some driig force; it udergoes chages to attai a state of ew equilibrium. Whateer is haeig to the system betwee these two equilibrium state is called a rocess. It may be rereseted by a ath which is the locus all the states i betwee o a - diagram as show i the figure below. For a system of gas i isto ad cylider arragemet which is i equilibrium, alterig ressure o the isto may be driig force which triggers a rocess show aboe i which the olume decreases ad ressure icreases. his haes util the icreasig ressure of the gas equalizes that of the surroudigs. If we locate the alues of all itermediate states, we get the ath o a - diagram. uasi-static rocess: uasi meas almost. A quasi-static rocess is also called a reersible rocess. his rocess is a successio of equilibrium states ad ifiite slowess is its characteristic feature. CYCLE Ay rocess or series of rocesses whose ed states are idetical is termed a cycle. Deartmet of Mechaical Egieerig, Page 7

8 BASIC EMODYNAMICS 0ME EMPEAUE he temerature is a thermal state of a body which distiguishes a hot body from a cold body. he temerature of a body is roortioal to the stored molecular eergy i.e., the aerage molecular kietic eergy of the molecules i a system. (A articular molecule does ot hae a temerature, it has eergy. he gas as a system has temerature). Istrumets for measurig ordiary temeratures are kow as thermometers ad those for measurig high temeratures are kow as yrometers. It has bee foud that a gas will ot occuy ay olume at a certai temerature. his temerature is kow as absolute zero temerature. he temeratures measured with absolute zero as basis are called absolute temeratures. Absolute temerature is stated i degrees cetigrade. he oit of absolute temerature is foud to occur at 7.5 C below the freezig oit of water. he: Absolute temerature = hermometer readig i C Absolute temerature is degree cetigrade is kow as degrees keli, deoted by K (SI uit). ZEO LAW OF EMODYNAMICS _ Zeroth law of thermodyamics states that if two systems are each equal i temerature to a third, they are equal i temerature to each other. Zeroth law of thermodyamics E EMOMEE AND EMOMEIC POPEY he zeroth law of thermodyamics roides the basis for the measuremet of temerature. It eables us to comare temeratures of two bodies ad with the hel of a third body ad say that the temerature of is the same as the temerature of without actually brigig ad i thermal cotact. I ractice, body i the zeroth law is called the thermometer. It is brought ito thermal equilibrium with a set of stadard temerature of a body, ad is thus calibrated. Later, whe ay other body is brought i thermal commuicatio with the thermometer, we say that the body has attaied equality of temerature with the thermometer, ad hece with body. his way, the body has the temerature of body gie for examle by, say the height of mercury colum i the thermometer. _ he height of mercury colum i a thermometer, therefore, becomes a thermometric roerty. here are other methods of temerature measuremet which utilize arious other roerties of materials, that are fuctios of temerature, as thermometric roerties. Six differet kids of thermometers, ad the ames of the corresodig thermometric roerties emloyed are gie below : Deartmet of Mechaical Egieerig, Page 8

9 BASIC EMODYNAMICS 0ME hermometer hermometric roerty. Costat olumes gas Pressure (). Costat ressure gas olume (). Alcohol or mercury-i-glass Legth (L) 4. Electric resistace esistace () 5. hermocoule Electromotie force (E) 6. adiatio (yrometer) Itesity of radiatio (I or J) Method i use before 954: _ Celsius ad Fahreheit scales are the two commoly used scales for the measuremet of temerature. Symbols C ad F are resectiely used to deote the readigs o these two scales. Util 954 the temerature scales were based o two fixed oits: (i) the steam oit (boilig oit of water at stadard atmosheric ressure), ad (ii) (ii) the ice oit (freezig oit of water). he fixed oits for these temerature scales are: emerature Celsius scale Fahreheit scale Steam oit 00 Ice oit 0 Iteral Method i use after 954: It was suggested by Keli that a sigle fixed oit oly was ecessary to establish a temerature. e oited out that trile oit of water (the state at which ice, liquid water ad water aour coexist i equilibrium) could be used as the sigle oit. he teth CGPM, i 954,adoted this fixed oit, ad alue was set at 0.0 C or 7.6 K i the Keli scale thus established. Corresodigly, the ice oit of 0 C o the Celsius scale becomes equal to 7.5 K o the Keli scale. Celsius ad Keli scales are distiguished by usig distict symbols t ad, the relatio betwee these two is the gie by: (K) = t( C) he Iteratioal Practical emerature Scale For the calibratio of thermometric istrumets the Seeth Geeral Coferece o Weight ad Measures held i 97 formulated a coeiet scale kow as the Iteratioal Practical emerature Scale. It was reised at hirteeth Geeral Coferece i 968. It cosists of reroducible referece temeratures or rimary fixed oits defied by a umber of ure substaces with assiged alues of temeratures determied with recisio o ideal or erfect gas temerature scale as gie i able Deartmet of Mechaical Egieerig, Page 9

10 BASIC EMODYNAMICS 0ME Fixed Poits of the Iteratioal Practical emerature Scale of 968 Equilibrium state Assiged alue of temerature It is stated here that : he trile oit reresets a equilibrium state betwee solid, liquid ad aour hases of a substace. Normal boilig oit is the temerature at which the substace boils at stadard atmosheric ressure of 760 mm g. Normal freezig oit is the solidificatio or the meltig oit temerature of the substace at stadard atmosheric ressure. Based o the aailable method of measuremet, the whole temerature scale may be diided ito four rages. he equatios for iterolatio for each rage are as follows:. From 59.4 C (trile oit of hydroge) to 0 C : A latium resistace thermometer of a stadard desig is used ad a olyomial of the followig form is fitted betwee the resistace of the wire t ad temerature t t = 0 ( + At + Bt + Ct) where 0 = resistace at the ice oit.. From 0 C to C (Atimoy oit) : It is also based o latium resistace thermometer. he diameter of the latium wire must lie betwee 0.05 ad 0. mm.. From C to C (Gold oit) : It is based o stadard latium ersus latium-rhodium thermocoule. Followig equatio betwee e.m.f. E ad temerature t is emloyed: E = a + bt + ct Deartmet of Mechaical Egieerig, Page 0

11 BASIC EMODYNAMICS 0ME ) A temerature scale of certai thermometer is gie by the relatio t = a l + b Where a ad b are costats ad is the thermometric roerty of the fluid i the thermometer.if at the ice oit ad steam oit the thermometric roerties are foud to be.5 ad 7.5 resectiely what will be the temerature corresodig to the thermometric roerty of.5 o Celsius scale. Solutio. t = a l + b...(gie) O Celsius scale : Ice oit = 0 C, ad Steam oit = 00 C From gie coditios, we hae 0 = a l.5 + b...(i) ad 00 = a l b...(ii) i.e., 0 = a b...(iii) ad 00 = a.05 + b...(i) Subtractig (iii) from (i), we get 00 =.6a or a = 6. Substitutig this alue i eq. (iii), we get b = = 5.8 Whe =.5 the alue of temerature is gie by t = 6. l (.5) 5.8= 5.6 C. ) A thermocoule with test juctio at t C o gas thermometer scale ad referece juctio at ice oit gies the e.m.f. as e = 0.0 t 5 0 4t m. he millioltmeter is calibrated at ice ad steam oits. What will be the readig o this thermometer where the gas thermometer reads 70 C? Deartmet of Mechaical Egieerig, Page

12 BASIC EMODYNAMICS 0ME UNI Work & eat Mechaics defiitio of work: Work is doe whe the oit of alicatio of a force moes i the directio of the force. he amout of work is equal to the roduct of the force ad the distace through which the oit of alicatio moes i the directio of the force. i.e., work is idetified oly whe a force moes its oit of alicatio through a obserable distace. Mathematically, W = F. dx oweer, whe treatig thermodyamics from a macroscoic oit of iew, it is adatageous to tie i the defiitio work with the cocets of systems, roerties ad rocesses. hermodyamic defiitio of work: It is a kid of iteractio that would occur at the system boudaries. It ca be ositie or egatie. Defiitio of Positie work is said to be doe by a system whe the sole effect exteral to the system could be reduced to the raisig of a weight. Commets: he word sole effect idicates that the raisig of weight should be the oly iteractio betwee the system ad surroudigs i order to say that there is work iteractio betwee the system ad the surroudigs. he hrase exteral to the system idicates that the work is a boudary heomeo. he magitude of work iteractio deeds uo the system boudary. his is illustrated with a examle. Figure : Equialece of Curret Work Iteractio betwee the System ad the Surroudigs For the two systems show i figure, system () comrisig battery aloe has work iteractio with the surroudigs, whereas for system () which icludes motor, weights etc alog with the battery, the work iteractio is zero. he word could be reduced to idicates that it is ot ecessary that weights should actually be raised i order to say that there is work iteractio betwee the system ad the Deartmet of Mechaical Egieerig, Page

13 BASIC EMODYNAMICS 0ME surroudigs. It is just sufficiet to hae a effect which is equialet to the raisig of weight. ere a electrical storage battery costitutes system whose termials are coected to a electrical resistace coil through a switch. he circuit exteral to the battery costitutes the surroudigs. Whe the switch is closed, the curret flow through the coil, ad the resistace (surroudigs) become warmer ad the charge of the battery (system) decreases. Obiously there has bee iteractio betwee the system ad the surroudigs. Accordig to mechaics this iteractio caot be classified as work because their has bee o actio of force through a distace or of torque through a agle. oweer, as er thermodyamics cocets, the battery (system) does work as the electrical eergy crosses the system boudary. Further, the electrical resistace ca be relaced by a ideal frictioless motor ulley arragemet which ca wid a strig ad thereby raise suseded weight. he sole effect, exteral to the system, is raisig of a weight. As such iteractio of battery with resistace coil is a work. Sig Coetios for work: Work is said to be ositie, if it is doe by the system o the surroudigs System Positie work Work is said to be egatie, if it is doe o the system by the surroudigs System Negatie work herefore, W system + W surroudigs = Zero he uit of work is N-m or Joule. he rate at which work is doe by, or uo, the system is kow as ower. he uit of ower is J/s or watt. Work is oe of the forms i which a system ad its surroudigs ca iteract with each other. here are arious tyes of work trasfer which ca get ioled betwee them. Work doe at the moig boudary of a system (Exressio for dislacemet work) dx Deartmet of Mechaical Egieerig, Page

14 BASIC EMODYNAMICS 0ME Cosider a isto-cylider arragemet which cotais certai workig fluid udergoig quasi-static rocess. Let = Pressure exerted by the fluid o the isto A = Area of c/s of the cylider dx = dislacemet of the isto whe the system has udergoe a ifiitesimal chage of state. Dislacemet work: dw = Force x dislacemet =.A x dx i.e., dw =.d Where d is the ifiitesimal chage i olume of the system. If the system udergoes a fiite chage of state from state () to state (). he the dislacemet work is gie by dw. d he itegratio of aboe equatio ca be doe oly if the relatioshi betwee P ad durig the rocess is kow i.e., if the ath of the rocess is well defied. ece, work is a ath fuctio. As work deeds o the ath of the rocess which it follows, there will be differet alues of work for differet rocess betwee two gie states. ece the differetials of the ath fuctios are i exact differetials. he symbol δ will be used to desigate iexact differetials. he magitude of the work trasfer by the system durig the rocess from state () to state () cotaiig uit mass of the fluid will be writte as, w W or W -. he rocess ca be rereseted by a full lie o a aroriate thermodyamic coordiate system (i this case - diagram) ad the area uder the cure gies the work doe by the system durig the rocess. A uasi-static B Process C d uasi-static d work Work-a ath fuctio Isectio of the diagram aboe shows that just by secifyig the ed states ad does ot determie the area (or work); the ature of the cure eeds to be kow. he cure may be arched uwards or it may sag dowwards, ad the area uder the cure will ary Deartmet of Mechaical Egieerig, Page 4

15 BASIC EMODYNAMICS 0ME accordigly. For the same iitial ad fial states, the work doe by the system i followig the aths A, B ad C are differet. herefore the work is a ath fuctio ad ot a oit fuctio. Accordigly the work trasfer across the system boudaries is ot classified as a thermodyamic roerty. he exressio w = d holds good uder the followig restrictios i) he system is closed ii) here is o frictio withi the system iii) he ressure ad all other roerties are the same o all the boudaries of the system i) he system is ot iflueced by motio, graity, caillarity, electricity ad magetism Exressio for Dislacemet work for arious uasi-static Processes (d work):. Costat olume rocess: (Isochoric Process). For a costat olume rocess i.e., = costat (d = 0 ) as rereseted i the - diagram below. W - = (W d ) - = 0.d but d = 0. Costat ressure rocess: (Isobaric rocess). For a closed system udergo a costat ressure rocess from state (olume ad ressure ) to a fial state (olume ). he rocess is rereseted i the - diagram as show below. = P W - Deartmet of Mechaical Egieerig, Page 5

16 BASIC EMODYNAMICS 0ME W - =.d, where = costat W - = d (W d ) - = ( ). yerbolic rocess i.e., = costat: he hyerbolic exasio rocess from state to state is rereseted o a - diagram as show below. W - =.d But = costat i.e., =, W - =.d. d Process i which = Costat = [l l ] where = Iitial ressure of the system i.e., (W d ) - = l = Iitial olume of the system = Fial ressure of the system = Fial olume of the system Deartmet of Mechaical Egieerig, Page 6

17 BASIC EMODYNAMICS 0ME 4. Polytroic rocess, i.e., = costat A olytroic rocess is rereseted o a - diagram as show below. W - =.d But = costat W - = = =. d -.d but = = (W d ) - = Figure: Process i which = Costat i.e., Where is called the idex of exasio or comressio Note:. Work is a trasiet heomeo i.e., it is reset oly durig a rocess. Mathematically seakig, work is a ath fuctio. dw = w w is wrog = w - i.e., w is iexact differetials.. For irreersible rocess w P.d Deartmet of Mechaical Egieerig, Page 7

18 BASIC EMODYNAMICS 0ME Other yes of Work rasfer. Shaft Work: Let Shaft work F t = agetial force o the shaft = adius of the shaft d = Agular dislacemet of the shaft i a iteral of time dt Shaft work i time iteral dt, is dw s = F t. AA Work doe / uit time = But ω = i.e., W s =.d N where N = rm of the shaft 60 Shaft work, W s = N watts 60 = F t..d dw s d. =.ω where ω = agular elocity, = orque dt dt. Stirrig Work: Stirrig work is othig but shaft work is doe o the system by usig a stirrer which is drie by a shaft. Figure: Paddle-wheel work Deartmet of Mechaical Egieerig, Page 8

19 BASIC EMODYNAMICS 0ME As the weight is lowered, ad the addle wheel turs, there is work trasfer ito the system which gets stirred. Sice the olume of the system remais costat, d = 0. If m is the mass of the weight lowered through a distace dz ad is the torque trasmitted by the shaft i rotatig through a agle d, the differetial work trasfer to the fluid is gie by w = mgdz = d ad the total work trasfer is w = lowered mgdz = W dz = d where W is the weight. Electrical Work: Whe a curret flows through a resistor, take as system, there is work trasfer ito the system. his is because the curret ca drie a motor, the motor ca drie a ulley ad the ulley ca raise a weight. he curret I, flows is gie by, I = d dc I where C = charge i coulombs = time i secods hus dc is the charge crossig a system boudary durig time d. If E is the oltage otetial, the work is w = E.dC w = = EI d EI d he electrical work is, w = lim δ 0 dw d his is the rate at which work is trasferred. = EI System boudary 4. Work doe i stretchig a wire: Cosider a wire as the system. If the legth of the wire i which there is a tesio Ŧ is chaged from L to L + dl, the ifiitesimal amout of work that is doe is equal to, w = - Ŧ dl he -e sig is used because a ositie alue of dl meas a exasio of the wire, for which work must be doe o the wire i.e., egatie work. I For a fiite chage of legth, w = - Ŧ dl Deartmet of Mechaical Egieerig, Page 9

20 BASIC EMODYNAMICS 0ME Withi the elastic limits, if E is the modulus of the elasticity, ad A is the cross sectioal area, the is the stress, ε is the strai, Ŧ = A = E.ε.A herefore w = - E.ε.AdL But dε = dl/l or dl = L x dε w = - Ŧ dl = - E.ε.A. L dε i.e., w = -EAL A L = - ε d ε 5. Work doe i chagig the area of a surface film: A film o the surface of a liquid has a surface tesio which is a roerty of the liquid ad the surroudigs. he surface tesio acts to make the surface area of the liquid a miimum. It has the uit of force er uit legth. he work doe o a homogeeous liquid film i chagig its surface area by a ifiitesimal amout da is w = - ζ da w = - ζ da whe ζ = surface tesio (N/m) 6. Magetizatio of a aramagetic field: he work doe er uit olume o a magetic material through which the magetic ad magetizatio fields are uiform is, i.e., w = - Where w = -.di.di = field stregth I = Comoet of the magetizatio field i the directio of the field. -e sig roides that a icrease i magetizatio (+e di) ioles -e work. Note: It may be oted i the aboe exressios that the work is equal to the itegral of the roduct of a itesie roerty ad the chage i its related extesie roerty. hese exressios are alid oly for ifiitesimally slow quasi-static rocess. Network rasfer: he etwork iteractio betwee the system ad the surroudigs for ay rocess will be the algebraic sum of all tyes of work iteractio that has take lace betwee the system ad the surroudigs. Deartmet of Mechaical Egieerig, Page 0

21 BASIC EMODYNAMICS 0ME herefore if W - reresets the et work trasfer the, W - = (W d ) - ± (W s ) - ± (W e ) - ± (W mag ) - ±... +e sig has to be used whe the work trasfer takes lace from the system to the surroudigs ad e sig to be used whe work trasfer is from the surroudigs to the system. eat: eat is a mode of eergy trasfer that takes lace betwee the system ad the surroudigs solely due to the temerature differece. hus, heat is a trasiet heomeo. It ca be recogized oly durig a rocess. It is ot a thermodyamic roerty of the system. Like work, heat is a ath fuctio i.e., the magitude of heat trasfer betwee the system ad surroudigs deeds uo the tye of rocess the system is udergoig. eat trasfer always takes lace from a regio of higher temerature to a regio of low temerature. he magitude of the heat trasfer ito uit mass of the fluid i the system durig a rocess from state () to state () will be writte as q q q q ad ot as reresets the total heat trasfer that takes lace whe the system udergoes a chage of state from state to state. Sig Coetio: eat trasfer is cosidered as ositie if it takes lace from the surroudigs to the system ad it is cosidered as egatie if it takes lace from the system to the surroudigs. Durig a adiabatic rocess, = 0 Uits: Sice heat is a form of eergy trasfer it will hae the same uits as that of eergy. I SI uits it is exressed i Joules (J) or Kilo Joules (kj). Comariso betwee work ad heat: Similarities: System +e System q q or q Both are ath fuctios ad iexact differetials. Both are boudary heomeo i.e., both are recogized at the boudaries of the system as they cross them. Both rereset trasiet heomeo; these eergy iteractios occur oly whe a system udergoes chage of state i.e., both are associated with a rocess, ot a state. Ulike roerties, work or heat has o meaig at a state. A system ossesses eergy, but ot work or heat. Cocets of heat ad work are associated ot with a store but with a rocess. -e Deartmet of Mechaical Egieerig, Page

22 BASIC EMODYNAMICS 0ME Dissimilarities: eat is eergy iteractio due to temerature differece oly; work is by reasos other tha temerature differece. I a stable system, there caot be work trasfer; howeer there is o restrictio for the trasfer of heat. he sole effect exteral to the system could be reduced to rise of a weight but i the case of a heat trasfer other effects are also obsered. eat is a low grade eergy whereas work is a high grade eergy.. A gas system, cofied by a isto ad cylider, udergoes a chage of state such that the roduct of ressure ad olume remais costat. If the rocess begis at a ressure of bar ad a olume 0.05m ad roceeds util the ressure falls to half its iitial alue, determie the magitude ad directio of the work flow. Solutio: = C i.e., hyerbolic rocess or = Gie, = x 0 5 Pa = 0.05 m =.5 x 0 5 m =? W - =? We hae, = = Dislacemet work, (W d ) - =.d = l = 9.6 J =.9 kj 5 x0 x0.05 = 0.0 m.5x0 = 5 Positie sig idicates work is doe by the system o the surroudigs.. A certai amout of gas is comressed from bar ad 0.m to 5 bar ad 0.0m. he rocess is accordig to the law = K. Determie the magitude ad directio of work. Solutio: Gie: = bar; = 0. m ; = 5 bar; = 0.0 We hae for a olytroic rocess, Dislacemet work, = (W d ) - = o fid the comressio idex, we hae, Deartmet of Mechaical Egieerig, Page

23 BASIC EMODYNAMICS 0ME C i.e., akig log s o both sides we hae l x0 (W d ) - =.l l l x i.e., work doe o the gas = kj = J = kj. A gas cofied i a cylider by a isto is at ressure of bar ad a olume of 0.05 m. he fial ressure is.5 bar. Determie the magitude ad directio of work trasfer for the followig rocesses. i) α, ii) α, iii) α ad i) α Solutio: Gie: = x 0 5 Pa; = 0.05 m ; =.5 x 0 5 Pa =? (W d ) =? i) α i.e., We hae, (W d ) - = we hae = =.d C. d.d (W d ) - =.5 = x0 [ ] = kj 0.05 = m -e sig idicates that work is doe o the system Deartmet of Mechaical Egieerig, Page

24 BASIC EMODYNAMICS 0ME ii) α i.e., = C As: (W d ) - =.9 kj iii) α (W d ) - = =.d i.e.,. d =. d = x C = 0.05 = kj -e sig idicates that work is doe o the system i) α = i.e., = C = = x We hae (W d ) - = 0.05 = 0.0 m.5.d =. d = d. = = = = m = Deartmet of Mechaical Egieerig, Page 4

25 BASIC EMODYNAMICS 0ME Substitutig the gie alues, we get (W d ) - =.6 kj +e sig idicates that work is doe by the system 4. he aerage heat trasfer from a erso to the surroudigs whe he is ot actiely workig is about 950 kj/hr. suose that i the auditorium cotaiig 000 eole the etilatio system fails. a) ow much does the iteral eergy of air i the auditorium icrease durig the first 5 miutes after the etilatio fails? b) Cosiderig the auditorium ad all the eole as system ad assumig o heat trasfer. to surroudigs, how much does the it. eergy of the system chage? ow do you accout for the fact that the temerature of air icreases? Solutio: a) Aerage heat trasfer er erso = 960 kj/hr = 960 / 60 = 5.8 kj /mi Aerage heat trasfer / erso for 5 mi = 7.5 kj Aerage heat trasfer for 5 mi i the auditorium cotaiig 000 eole = 7.5 x 000 = 7500 kj/mi From first law of D, we hae = E + W 7500 = E + 0 E = 7.5 MJ Deartmet of Mechaical Egieerig, Page 5

26 BASIC EMODYNAMICS 0ME UNI First Law of hermodyamics to oe system: First Law of hermodyamics I the case of closed system there is oly eergy trasfer across the system boudary. But i may egieerig alicatios we come across oe systems where i both mass ad eergy trasfer takes lace. he eergies that cross the system boudary are as follows. ) Iteral eergy: Each kg of matter has the iteral eergy u ad as the matter crosses the system boudary the eergy of the system chages by u for eery kg mass of the matter that crosses the system boudary. ) Kietic eergy: Sice the matter that crosses the system boudary will hae some elocity say each kg of matter carries a k.e. / thus causig the eergy of the system to chage by this amout for eery kg of matter eterig the system boudary. ) Potetial eergy: P.E. is measured with referece to some base. hus Z is the eleatio of the matter that is crossig the system boudary, the each kg of matter will ossess a P.E. of gz. 4) Flow eergy or Flow work: his eergy is ot directly associated with the matter crossig the system boudary. But it is associated with the fact that there must be some umig rocess which is resosible for the moemet of the matter across the system boudary. hus exteral to the system there must be some force which forces the matter across the system boudary ad the eergy associated with this is called flow eergy. Flow Work: Cosider a flow rocess i which a fluid of mass dm is ushed ito the system at sectio ad a mass dm is forced out of the system at sectio as show i fig. A dm, dm F F dl dl I order to force the fluid to flow across the boudary of the system agaist a ressure, work is doe o the boudary of the system. he amout of work doe is δw = - F.dl, Deartmet of Mechaical Egieerig, Page 6

27 BASIC EMODYNAMICS 0ME Where F is the force ad dl is the ifiitesimal dislacemet, but F = A δw = - A dl = - d i.e., the flow work at sectio = - Similarly, the work doe by the system to force the fluid out of the system at sectio = + ece et flow work = For uit mass, the flow work is ( ). Flow work is exressed etirely i terms roerties of the system. he et flow work deeds out o the ed state of the fluid ad it is a thermodyamics roerty. Also the fluid cotais eergies like iteral eergy, otetial eergy ad due to the motio of the fluid, kietic eergy, i additio to the flow work. Whe a fluid eters a oe system, these roerties will be carried ito the system. Similarly whe the fluid leaes the system, it carries these eergies out of the system. hus i a oe system, there is a chage i eergy of the system. 5. Cotrol olume: he first ad most imortat ste i the aalysis of a oe system is to imagie a certai regio eclosig the system. his regio haig imagiary boudary is called cotrol olume, which ca be defied as follows. A C.. is ay olume of fixed shae, ad of fixed ositio ad orietatio relatie to the obserer. Across the boudaries of the C.. aart from mass flow, eergy trasfer i the form of heat ad work ca take lace just as similar to the eergy trasfer across the boudaries of a system. hus the differece betwee C.. ad system are i) he system boudary may ad usually does chage shae, ositio, orietatio relatie to the obserer. he C.. does ot by defiitio. ii) Matter may ad usually does flow across the system boudary of the C.. No such flow takes lace across the system boudary by defiitio. First law of thermodyamics for a oe system (Flow rocess): We hae st law of thermodyamics to a closed system as, δ δw = du + d(ke) + d (PE) = d [E] he subscrit O refers to the states of the system withi the boudary. I the case of oe system, eergy is trasferred ito & out of the system ot oly by heat ad work but also by the fluid that eters ito ad leaes the boudary of the system i the form of iteral eergy, graitatioal otetial eergy, kietic eergy i additio to the eergy i the flow work. hus, whe the first law is alied to a oe system, the eergy eterig ito the system must be equal to the eergy leaig the system i additio to ay accumulatio of eergy withi the system. δ d[e] δ Deartmet of Mechaical Egieerig, Page 7

28 BASIC EMODYNAMICS 0ME dm dm u u gz gz he flow rocess is show i fig. his aalysis ca be exressed mathematically as, W dm u gz dm u gz d E () Where state () is the eterig coditio ad state () is the leaig coditio of the fluid. his is a geeral equatio of the first law of thermodyamics alied to oe system. Note: he equatio is alid to both oe ad closed system. For closed system, dm =0 & dm =0 Eergy Equatio for oe system: he geeral form of first law of thermodyamics alied to a oe system is called steady-flow eergy equatio (SFEE) i.e., the rate at which the fluid flows through the C.. is costat or steady flow. SFEE is deeloed o the basis of the followig assumtios. i) he mass flow rate through the C.. is costat, i.e., mass eterig the C.. / uit time = mass leaig the C.. /uit time. his imlies that mass withi the C.. does ot chage. ii) he state ad eergy of a fluid at the etrace ad exit do ot ary with time, i.e., there is o chage i eergy withi the C.. iii) he rates of heat ad work trasfer ito or out of the C.. do ot ary with time. For a steady flow rocess, m m m & d(e) 0 = 0 as f () & W f () SFEE o the basis uit mass: Eergy eterig to the system = eergy leaig the system Deartmet of Mechaical Egieerig, Page 8

29 BASIC EMODYNAMICS 0ME i.e., u gz W u gz or h gz W h gz or W h gz Where = heat trasfer across the C., W = shaft work across the C., h = Ethaly, = elocity, Z = eleatio ad g = graitatioal acceleratio Dislacemet work for a flow rocess (oe system): From SFEE, whe chages i kietic & otetial eergies are eglected, δq δw = dh Or δw = δq dh --- () From the st law of thermodyamics, we hae δq δw = du For a re. rocess, δw = Pd δq = du + Pd Also, from the defiitio of ethaly, h = u + Or dh = du + d () Sub δq & dh i equatio (i) W = - d δw = [du +.d] [du + d ()] =.d.d.d Note: With egligible PE & KE, for a o-flow re. rocess, the work iteractio is equal to where as for a steady-flow re. rocess, it is equal to Alicatio of SFEE: d. d i) Nozzle ad Diffuser: Nozzle is a duct of aryig c/s area i which the elocity icreases with a corresodig dro i ressure. Sice the flow through the ozzle occurs at a ery high seed, there is hardly ay time for a fluid to gai or loose heat ad hece flow of the fluid is assumed to be adiabatic. Ad also there is o work iteractio durig the rocess, i.e., W s = 0, = 0, Z = Z Deartmet of Mechaical Egieerig, Page 9

30 BASIC EMODYNAMICS 0ME We hae from SFEE, W = h + PE + KE 0 = h h + h h urbie ad Comressor (rotary): urbie is a deice which roduces work by exadig a high ressure fluid to a low ressure. he fluid is first accelerated i a set of ozzle ad the directed through cured moig blades which are fixed o the rotor shaft. he directio of the fluid chages which it flows through the moig blades, due to which there is a chage i mometum ad a force exerted o the blades roducig torque o the rotor shaft. Sice the elocity of flow of the fluid through the turbie is ery high, the flow rocess is geerally assumed to be adiabatic, hece heat trasfer q = 0. he chage i PE is eglected as it is egligible. SFEE is W - = (h h ) ½ ( ) If mass flow rate is m, the, W Z Z = Z m h h m W - Z Z = Z Watts Comressor is a deice i which work is doe o the fluid to raise its ressure. A rotary comressor ca be regarded as a reersed turbie. Sice work is doe o the system, the rate of work i the aboe equatio is egatie ad the ethaly after comressio h will be greater tha the ethaly before comressio h. Deartmet of Mechaical Egieerig, Page 0

31 BASIC EMODYNAMICS 0ME ii) hrottlig Process: Whe a fluid steadily through a restricted assages like a artially closed ale, orifice, orous lug etc., the ressure of the fluid dros substatially ad the rocess is called throttlig. I a throttlig rocess, exasio of the fluid takes lace so raidly that o heat trasfer is ossible betwee the system ad the surroudigs. ece the rocess is assumed to occur adiabatically. he work trasfer i this rocess is zero. SFEE is - W - = h + KE + PE We hae, = 0; W = 0; Z = Z, i.e., h = h I a throttlig rocess, the ethaly remais costat. he throttlig rocess is irreersible because whe a fluid is throttled, it asses through a series of o-equilibrium states. iii) eat Exchager: A heat exchager is a deice i which heat is trasferred from oe fluid to aother. It is used to add or reduced heat eergy of the fluid flowig through the deice. adiator i a automobile, codeser i a steam ower ad refrigeratio lats, eaorator i a refrigerator are examles of heat exchagers. here will be o work iteractio durig the flow of the fluid through ay heat exchager. Eg: i) Steam codeser: Used to codese the steam. It a deice i which steam loses heat as it asses oer the tubes through which water is flowig. Figure: eat Exchager We hae KE = 0, PE = 0 (as their alues are ery small comared to ethalies) W = 0 (sice either ay work is deeloed or absorbed) SFEE is = h h i.e., h = h --- () Where = heat lost by kg of steam assig through the codeser. Deartmet of Mechaical Egieerig, Page

32 BASIC EMODYNAMICS 0ME Assumig there are o other heat iteractios excet the heat trasfer betwee steam ad water, the = heat gaied by water assig through the codeser. m w h h m C w w w w w w Substitutig i the aboe equatio (), Where m w h - h m w C w ( - w w = mass of coolig water assig through the codeser C w = secific heat of water ) I a codeser there are steady flow streams amely (i) aour that losses heat (ii) he coolat (water) that receies heat. Let m m w s = mass flow rate of coolat = mass flow rate of steam h w = Ethaly-coolat etry h s = Ethaly-steam etry h w, h s = Ethaly of coolat, steam at exit m w h w + s or m m w s m h s = h h s w m w h w + s h s h h s w m ii) Eaorator: A eaorator is a comoet of a refrigeratio system ad is used to extract heat from the chamber which is to be ket at low temerature. he refrigeratig liquid eters the eaorator, absorbs latet heat from the chamber at costat ressure ad comes out as a aour. SFEE is m h + = m h Sice W = 0, KE = PE = 0 = m (h h ) is take as ositie because heat flows from the chamber to the eaorator coil. Deartmet of Mechaical Egieerig, Page

33 BASIC EMODYNAMICS 0ME Figure: Eaorator i) Boiler: It is a equimet used for the geeratio of the steam. hermal eergy released by combustio of fuel is trasferred to water which aourizes ad gets coerted ito steam at the desired ressure ad temerature. he steam thus geerated is used for a. Producig mechaical work by exadig it i steam egie or steam turbie. b. eatig the residetial ad idustrial buildigs i cold weather ad c. Performig certai rocesses i the sugar mills, chemical ad textile idustries.. elocity chage is egligible =. Chage i eleatio is also egligible Z = Z. Work doe = 0 SFEE is h + q = h q = h h = (u u ) + ( ) First law of thermodyamics for a closed system udergoig a cyclic rocess Whe a system udergoes a thermodyamic cyclic rocess, the the et heat sulied to the system from the surroudigs is equal to the et work doe by the system o its surroudig. i.e., = W where reresets the sum for a comlete cycle. he first law of thermodyamics caot be roed aalytically, but exerimetal eidece has reeatedly cofirms its alidity ad sice o heomeo has bee show to cotradict it, therefore the first law is acceted as a law of ature. Joule s Exerimet: Deartmet of Mechaical Egieerig, Page

34 BASIC EMODYNAMICS 0ME Figure: Joule s Exerimet Figure: Cycle comleted by a system with two eergy iteractios i.e., work trasfer followed by heat trasfer Figure shows the exerimet for checkig the first law of thermodyamics. he work iut to the addle wheel is measured by the fall of weight, while the corresodig temerature rise of liquid i the isulated cotaier is measured by the thermometer. he rocess - udergoe by the system is show i figure i.e., W -. Let the isulatio be remoed. he system ad the surroudig iteract by heat trasfer till the system returs to its origial temerature, attaiig the coditio of thermal equilibrium with the atmoshere. he amout of heat trasfer - from the system durig this rocess - is show i figure. he system thus executes a cycle, which cosists of a defiite amout of work iut W - to the system followed by the trasfer of a amout of heat - from the system. Joule carried out may such exerimets with differet tye of work iteractios i a ariety of systems, he foud that the et work iut the fluid system was always roortioal to the et heat trasferred from the system regardless of work iteractio. Based o this exerimetal eidece Joule stated that, Whe a system (closed system) is udergoig a cyclic rocess, the et heat trasfer to the system is directly roortioal to the et work doe by the system. his statemet is referred to as the first law for a closed system udergoig a cyclic rocess. i.e., W If both heat trasfer ad work trasfer are exressed i same uits as i the S.I. uits the the costat of roortioality i the aboe equatio will be uity ad hece the mathematical form of first law for a system udergoig a cyclic rocess ca be writte as i.e., = W If the cycle ioles may more heat ad work quatities as show i figure, the same result will be foud. Deartmet of Mechaical Egieerig, Page 4

35 BASIC EMODYNAMICS 0ME Figure: Cyclic Process o a Proerty Diagram For this cyclic rocess the statemet of first law ca be writte as 4 W 4 he cyclic itegral i the aboe equatio ca be slit ito a series of o cyclic itegral as 4 or = W + W + W W i.e., = W or ( ) cycle = ( W) cycle 4 W his is the first law for a closed system udergoig a cyclic rocess. i.e., it is stated as W Whe a closed system is udergoig a cyclic rocess the algebraic sum of heat trasfers is equal to the algebraic sum of the work trasfers. First law for a closed system udergoig a o-cyclic rocess (i.e., for a chage of state): If a system udergoes a chage of state durig which both heat trasfer ad work trasfer are ioled, the et eergy trasfer will be stored or accumulated withi the system. If is the amout of heat trasferred to the system ad W is the amout of work trasferred from the system durig the rocess as show i figure, W 4 W System he et eergy trasfer (-W) will be stored i the system. Eergy i storage is either heat or work ad is gie the ame iteral eergy or simly, the eergy of the system. 4 W -W = E or = E + W Deartmet of Mechaical Egieerig, Page 5

36 BASIC EMODYNAMICS 0ME If there are more eergy trasfer quatities ioled i the rocess as show i figure. First law gies ( + ) = E + (W + W W W 4 ) i.e., eergy is thus cosered i the oeratio. herefore the first law is a articular formulatio of the ricile of the coseratio of eergy. It ca be show that the eergy has a defiite alue at eery state of a system ad is therefore, a roerty of a system. Eergy A roerty of the system: Figure: First law to a o cyclic rocess Cosider a system that udergoes a cycle, chagig from state to state by rocess A ad returig from state to state by rocess B. We hae from st law of thermodyamics, W For the rocess, -A--B-, W A B A B Cosiderig the two searate rocesses, we hae A W System W W W 4 B A W B W --- () Now cosider aother cycle, the system chagig from state to state by rocess A, as before ad returig to state by rocess C. For this cycle we ca write A C A W C W --- () Deartmet of Mechaical Egieerig, Page 6

37 BASIC EMODYNAMICS 0ME Subtractig () from (), we get B C B W C W Or, by rearragig, B W C W Sice B ad C rereset arbitrary rocesses betwee state to state, we coclude that the quatity ( - W) is the same for all rocesses betwee state ad state. ( - W) deeds oly o the iitial ad fial states ad ot o the ath followed betwee the two states. his is a oit fuctio ad differetial is a roerty of the system. his roerty is called the eergy of the system, E. herefore, we ca write - W = de Or = de + W If it is itegrated betwee iitial ad fial states, ad, we get - = E E + W - i.e., - - W - = E E he aboe equatio is the statemet of first law for a closed system udergoig a o cyclic rocess, where - reresets the et heat trasfer betwee the system ad the surroudigs durig the rocess, W - reresets et work trasfer betwee the system ad the surroudigs durig the rocess ad (E E ) reresets the chage i the eergy of the system durig the rocess. Classificatio of Eergy of the System: he eergy E is a extesie roerty ad the secific eergy e = E/m (J/kg) is a itesie roerty. Eergy E reresets the total eergy of the system. i.e., E = kietic eergy (KE) + Potetial Eergy (PE) + remaiig forms of eergy. Sice K.E ad P.E are macroscoic quatities ad ca be measured ery easily ad so they are cosidered searately i thermodyamics. he remaiig eergies (associated with the motio ad ositio of the molecules, eergy associated with the structure of the atom, chemical eergy etc), which caot be measured directly ad is the summatio of all microscoic eergies is called iteral eergy of the system. Iteral eergy: It is the eergy associated with iteral structure of matter. his eergy ca ot be determied i its absolute alues. But it is ossible to determie the chage i iteral eergy of the system udergoig a rocess by first law of thermodyamics. otal eergy E = KE + PE + IE Sice the terms comrisig E are oit fuctios, we ca write Deartmet of Mechaical Egieerig, Page 7

38 BASIC EMODYNAMICS 0ME de = d(ke) + d (PE) + du he first law of thermodyamics for a chage of state of a system may therefore be writte as = du + d (KE) + d (PE) + W I words this equatio states that as a system udergoes a chage of state, eergy may cross the boudary as either heat or work, ad each may be ositie or egatie. he et chage i the eergy of the system will be exactly equal to the et eergy that crosses the boudary of the system. he eergy of the system may chage i ay of three ways, amely, by a chage i IE, KE or P.E Sub. For KE ad PE i the aboe equatio = du + d( m ) + d (mgz) + W I the itegral form this equatio is, assumig g is a costat - = U U + m + mg (Z Z ) + W - I most of the situatios the chages i KE ad PE are ery small, whe comared with the chages i iteral eergies. hus KE ad PE chages ca be eglected. = du + W Law of coseratio of eergy (d corollary of first law of thermodyamics) From first law of thermodyamics - = E E + W - his equatio i effect, a statemet of the coseratio of eergy. he et chage of the eergy of the system is always equal to the et trasfer of eergy across the system boudary as heat ad work. For a isolated system, = 0, W = 0 E E = 0 For a isolated system, the eergy of the system remais costat. herefore, the first law of thermodyamics may also be stated as follows, eat ad work are mutually coertible but sice eergy ca either be created or destroyed, the total eergy associated with a eergy coersio remais costat. Peretual Machie of first kid ( rd Corollary): Ay system which iolates the first law of thermodyamics is called the Peretual Motio machie of first kid. i.e., It is imossible to costruct a eretual motio machie of first kid. A eretual machie is oe which ca do cotiuous work without receiig eergy from other systems or surroudigs. It will create eergy o its ow ad thus iolates first law. But from our exeriece we also kow that it is imossible to costruct such a machie, as frictioal resistace would ot allow it to ru for a idefiite eriod. Deartmet of Mechaical Egieerig, Page 8

39 BASIC EMODYNAMICS 0ME Problems:. I a cyclic rocess, heat temerature are kj, -5. kj, -.56 kj ad +.5 kj. What is the et work for this cyclic rocess. Solutio: st law of thermodyamics for a cyclic rocess is W i.e., Net work = = 7.44 kj. Cosider a cyclic rocess i a closed system which icludes three heat iteractios, amely = 0 kj, = -6kJ, ad = -4 kj ad two work iteractios for which W = 4500 N-m. Comute the magitude of the secod work iteractio W i Nm. Solutio: We hae for a closed system udergoig cyclic rocess, W = W W = 5500 Nm. Whe the state of a system chages from state to state alog the ath -- as show i figure, 80 kj of heat flows ito the system ad the system does 0 kj of work. (a) ow much heat flows ito the system alog the ath -4- if work doe by the system is 0 kj (b) whe the state of the system is retured from state to state alog the cured ath, the work doe o the system is 0 kj. Does the system absorb or liberate heat? Fid its magitude. (c) If U = 0 ad U 4 = 40kJ, fid the heat absorbed i the rocess -4 ad 4- resectiely. Solutio: a) Alog the ath --, From st law of thermodyamics, - = U -U + W - From the data gie, 80 = (U U ) + 0 (U U ) = 50 kj A 4 Alog the ath -4-, we hae - = U U + W - From the data gie, - = Deartmet of Mechaical Egieerig, Page 9

40 BASIC EMODYNAMICS 0ME = 60 kj i.e., Work is doe by the system b) Alog the ath -A-, (U U ) = - W - Or - = (U U ) + W - = = -70 kj Negatie sig idicates that heat is liberated from the system. c) Alog the ath -4-4 = U 4 U + W -4 = (sice W -4- = W -4 + W 4- = = 0) = 50 kj Positie sig idicates heat is absorbed by the system Alog the ath 4-4- = U U 4 + W 4- = = 0 kj 4. A domestic refrigerator is loaded with food ad the door closed. Durig a certai eriod the machie cosumes kwhr of eergy ad the iteral eergy of the system dros by 5000 kj. Fid the et heat trasfer. for the system. Solutio: W - = kwhr = - x600 kj U U = kj From st law, - = (U -U ) + W - = = kj = mj 5. For the followig rocess i a closed system fid the missig data (all i kj) Process W U U U a) b) c) d) Solutio: Process (a): = U + W Deartmet of Mechaical Egieerig, Page 40

41 BASIC EMODYNAMICS 0ME = U U + W - but U -U = 5 U = 5 = = 5 kj Process (b): = U U + W 5 = -6-U -6 7 = -U U = -7 kj U = U U = = kj Process (c) - 7 = U U = kj Process (d) U = U U = - 0 U = -0 = -7 kj = 8 U = - 0 U = 8 kj A = = - 7 kj 6. A fluid system, cotaied i a isto ad cylider machie, asses through a comlete cycle of four rocesses. he sum of all heat trasferred durig a cycle is -40 kj. he system comletes 00 cycles miutes. Comlete the followig table showig the method for each item, ad comute the et rate of work outut i kw. Process (kj/mi) W (kj/mi) E (kj/mi) Solutio: Gie cycle = -40 kj, Process -: - = (E E ) + W - No. of cycle = 00 cycles / mi 0 = E + W - E = -440 kj/mi Process -: 4000 = E = 4000 kj/mi Process -4: -400 = W -4 W -4 = kj/mi Deartmet of Mechaical Egieerig, Page 4

42 BASIC EMODYNAMICS 0ME Process 4-: = -40 kj cycle he system comletes 00 cycle/mi = -40 x 00 cycle = kj / mi But, = = = kj/mi Also, de = 0, sice cyclic itegral of ay roerty is zero ( E) - + ( E) - + ( E) -4 + ( E) 4- = ( E) 4- =0 ( E) 4- = 5540 kj/mi herefore 4- = ( E) 4- + W 4- Sice cycle = W 4- W 4- = -440 kj/mi W cycle = kj/mi ate of work outut = =. kw 60 Deartmet of Mechaical Egieerig, Page 4

43 BASIC EMODYNAMICS 0ME UNI 4 Secod law of hermodyamics he first law states that whe a closed system udergoes a cyclic rocess, the cyclic itegral of the heat is equal to the cyclic itegral of the work. It laces o restrictios o the directio of the heat ad the work. As o restrictios are imosed o the directio i which the rocess may roceed, the cycle may be reersed ad it will ot iolate the first law. Examle (): A closed system that udergoes a cycle iolig work ad heat. I the examle cosidered the system udergoes a cycle i which work is first doe o the system by the addle wheel as the weight is lowered. he let the cycle be comleted by trasferrig heat to the surroudig. From exeriece it has bee leart that we caot reerse this cycle. i.e., if we trasfer heat to the gas, as show by the dotted lie, the temerature of the gas will icrease, but the addle wheel will ot tur ad lift the weigh. his system ca oerate i a cycle i which the heat ad work trasfers are both egatie, but it caot oerate i a cycle whe both are ositie, ee though this would ot iolate the first law Examle (); igh emerature Low emerature Deartmet of Mechaical Egieerig, Page 4

44 BASIC EMODYNAMICS 0ME Let two systems, oe at a high temerature ad the other at a low temerature udergoes a rocess i which a quatity of heat is trasferred from the high temerature system to the low temerature system. From exeriece we kow that this rocess ca take lace. But the reerse rocess i which heat is trasferred from the low temerature system to the high temerature system does ot occur ad that it is imossible to comlete the cycle by heat trasfer oly. hese two examles lead us to the cosideratio of the heat egie ad heat um (i.e., refrigerator). Exeriece tells us that the reersed rocesses described aboe do ot hae. he total eergy of each system would remai costat i the reersed rocess ad thus there would be o iolatio of the first law. It follows that there must be some other atural ricile i additio to the first law ad ot deducible from it, which goers the directio i which a rocess ca take lace i a isolated system. his ricile is the Secod law of thermodyamics. PEPEUAL MOION MACINE OF FIS KIND (PMMKI) No machie ca roduce eergy without corresodig exediture of eergy without corresodig exediture of eergy i.e., it is imossible to costruct a PMMK of first kid. he machie iolates the first law of thermodyamics. All attemts made so for to make PMMKI hae failed, thus showig the alidity of the first law. Secod law of hermodyamics here are two classical statemets of the secod law of thermodyamics ) Keli Plack statemet ) Clausius statemet Keli Plack statemet It is imossible to costruct a deice which will oerate i a cycle & roduce o effect other tha the raisig of a weight ad the exchage of heat with a sigle reseroir No actual or ideal egie oeratig i cycles ca coert ito work all the heat sulied to the workig substace, it must discharge some heat ito a aturally accessible sik because of this asect ad the secod law is ofte referred as the law of degradatio of eergy. Deartmet of Mechaical Egieerig, Page 44

45 BASIC EMODYNAMICS 0ME A directioal imlicatio of the d Law PEPEUAL MOION MACINE OF SECOND KIND (PMMKII) Clausius Statemet Without iolatig the first law a machie ca be imagied which would cotiuously absorb heat from a sigle thermal reseroir ad would coert this heat comletely ito work. he efficiecy of such a machie would be 00%. his machie is called PMMK II. A machie of this kid will iolate the secod law of thermodyamics ad hece does ot exist. It is imossible to costruct a heat um which oeratig i a cycle will roduce o effect other tha the trasfer of heat from a low temerature thermal reseroir to a higher temerature thermal reseroir. hat is i order to trasfer heat from a low temerature thermal reseroir to a high temerature thermal reseroir work must be doe o the system by the surroudigs. Although the Keli Plack ad Clausius statemets aear to be differet, they are really equialet i the sese that a iolatio of oe statemet ioles iolatio of the other. Deartmet of Mechaical Egieerig, Page 45

46 BASIC EMODYNAMICS 0ME Proof of iolatio of the Keli Plak statemet results i iolatio of the Clausius statemet. Cosider a heat egie that is oeratig i a cyclic rocess takes heat ( ) from a high temerature reseroir & coerts comletely ito work (W), iolatig the Keli Plack statemet. Let the work W, which is equal to, be utilized to drie a heat um as show. Let the heat um take i L amout of heat from a low temerature reseroir ad um ( + L ) amout of heat to the high temerature reseroir. From the diagrams we see that a art of heat, umed to the high temerature reseroir is deliered to the heat egie, while there remais a heat flow L, from the low temerature reseroir to the high temerature reseroir, which i fact iolates the clausius statemet. Proof of iolatio of the Clausius statemet results i iolatio of the Keli Plack statemet. Cosider a heat um that oeratig i a cyclic rocess takes i a amout of heat L from L ad trasfer the heat equialet amout of heat L to the iolatig the Clausius statemet. Deartmet of Mechaical Egieerig, Page 46

47 BASIC EMODYNAMICS 0ME Let a amout of heat, which is greater tha L, be trasferred from high temerature reseroir to a heat egie, a amout of heat L, be rejected by it to the L ad a amout of work W which is equal to ( L ) be doe by the heat egie o the surroudig. Sice there is o chage i heat trasfer i the L, the heat um, the ad the heat egie together ca be cosidered as a deice which absorbs a amout of heat ( L ) from the ad roduce a equal amout of work W = L which i fact iolates the Keli Plaks statemet. eersibility ad Irreersibility If 00% efficiecy is uattaiable, what is the max ossible efficiecy which ca be attaied ad what factors romote the attaimet of this max alue? I tryig to aswer these questios, thermodyamics has ieted & used the cocet of reersibility, absolute temerature ad etroy. eersible Process: -for a system is defied as a rocess which oce haig take lace, ca be reersed ad leaes o chage i either the system or surroudigs. Oly ideal rocesses ca do this ad restore both system ad surroudigs to their iitial states. ece a ideal rocess must be a reersible rocess. No real rocess is truly reersible but some rocesses may aroach reersibility, to a close aroximatio. Examle: ) Frictioless relatie motio ) Extesio ad comressio of a srig ) Frictioless adiabatic exasio or comressio of fluid. 4) Polytroic exasio or comressio etc., he coditios for a rocess to be reersible may be gie as follows: i) here should be o frictio ii) here should be o heat trasfer across fiite temerature differece. iii) Both the system ad surroudig be stored to origial state after the rocess is reersed. Ay rocess which is ot reersible is irreersible. Examle: Moemet of solids with frictio, A flow of iscous fluid i ies ad assages mixig of two differet substaces, A combustio rocess. Eery quasistatic rocess is reersible, because a quasistatic rocess is of a ifiite successio of equilibrium states. Deartmet of Mechaical Egieerig, Page 47

48 BASIC EMODYNAMICS 0ME Examles of reersible rocesses: (a) Frictioless relatie motio Examles of Irreersible rocesses: (i) Solid Frictio Proof that heat trasfer through a fiite temerature differece is irreersible (b) Extesio of a srig (ii) Free exasio eat trasfer through a fiite temerature differece eat trasfer through a fiite temerature differece is irreersible Deartmet of Mechaical Egieerig, Page 48

49 BASIC EMODYNAMICS 0ME Proof that Urestraied exasio makes rocess irreersible he CANO cycle: Carot was the first ma to itroduce the cocet of reersible cycle. he CANO egie works betwee & L. igh tem reseroir CANO Egie INIAL SAE L Low temerature reseroir W EESED POCESS he Carot cycle cosists of a alterate series of two reersible isothermal ad two reersible adiabatic rocesses. Sice the rocesses i the cycle are all reersible the Carot cycle as a whole is reersible. he Carot cycle is ideedet of the ature of the workig substace ad it ca work with ay substace like gas, aour, electric cell etc., CANO CYCLE ENGINE CANO CYCLE Deartmet of Mechaical Egieerig, Page 49

50 BASIC EMODYNAMICS 0ME ) Process : Gas exads isothermally absorbig heat from the source at emerature. Work doe durig this rocess is gie by the area uder (W ) ) Process : Durig this rocess cylider is thermally isolated from the heat reseroir ad the head is isulated by the iece of erfect isulator. Gas exads reersibly ad adiabatically to temerature to oit. Work doe is W. ) Process 4: Cylider is i cotact with the heat reseroir at. Gas is isothermally ad reersibly comressed to oit 4 rejectig a amout of heat to the sik. he work doe o the W 4. 4) Process 4 : Cylider is agai isolated thermally from the thermal reseroir; gas is recomressed adiabatically ad reersibly to oit. he cycle is ow comlete. Work doe is W 4 he efficiecy of the Carot egie is gie by, herefore, η η carot Work outut eat iut eat added durig rocess eat rejected carot Sice rocess Process 4 γ durig rocess m l m m l (5) m l 4 l isreersible adiabatic 4 η carot isalso reersible adiabatic γ, herefore, 4 or m l 4 (7) 4 4 () (6) () () Low emerature η carot (8) igh emerature Deartmet of Mechaical Egieerig, Page 50

51 BASIC EMODYNAMICS 0ME From the aboe equatio we ca hae the followig coclusios. Ee i a ideal cycle, it is imossible to coert all the eergy receied as heat from the source ito mechaical work. We hae to reject some of the eergy as heat to a receier at a lower temerature tha the source (sik). Sice Carot cycle cosists of reersible rocesses, it may be erformed i either directio. Process - 4 Process 4 - Process - Process - COP of heat um = eersible adiabatic exasio eersible isothermal exasio eersible adiabatic Comressio eersible isothermal Comressio W L L emerature falls dow from to L emerature remais costat emerature icreases from L to emerature remais costat COP of refrigerator = L W L igh tem reseroir L CANO eat Pum L Low temerature reseroir L L W L Carot heat um with a gas Deartmet of Mechaical Egieerig, Page 5

52 BASIC EMODYNAMICS 0ME Problems: A heat egie works o the Carot cycle betwee temerature 900 C & 00 C. If the egie receies heat at the higher temerature at the rate of 60 kw, calculate the ower of the egie. Also, th. = = 7 k L = = 47 k th. L W W. W. W = x 60 = 5.8 kw CANO theorem ad corollary. th A irreersible heat egie ca ot hae a efficiecy greater tha a reersible oe oeratig betwee the gie two temeratures. Fig. (a) Fig. (b) I other words, for the assumed coditio that I is more efficiet tha, we fid that heat is beig moed cotiuously from L to without the exteral aid. Deartmet of Mechaical Egieerig, Page 5

53 BASIC EMODYNAMICS 0ME Istead of simly moig the heat as show i fig (b), we could direct the flow of eergy from the reersible egie directly ito the irreersible egie, as i fig (c), whose efficiecy is 50% would allow to drie egie, ad at the same time delier 0 KJ of work to somethig outside of the system. his meas the system exchages heat with a sigle reseroir ad deliers work. hese eets hae eer bee kow to hae. Corollary We say that the assumtio that I is more efficiet tha is imossible. I All reersible egies hae the same efficiecy whe workig betwee the same two temeratures. Cosider two reersible egies ad, oeratig betwee the two temeratures. If we imagie driig backward, the Carot theorem states that. If dries backward, the It therefore follows that If this were ot so, the more efficiet egie could be used to ru the less efficiet egie i the reerse directio ad the et result would the trasfer of heat from a body at low temerature to a high temerature body. his is imossible accordig to the secod law. Deartmet of Mechaical Egieerig, Page 5

54 BASIC EMODYNAMICS 0ME Suose & are two reersible egies workig betwee the two same reseroirs as show let us assume that is more efficiet tha. By our assumtio W W i.e., i.e., L L L L & W W =. () L L Now let egie be reersed so that it abstracts heat from L at L ad deliers heat to at. Sice the heat required by is also we ca relace the reseroir by a coductor betwee &. his ew combiatio would become a PMMK II because it would abstract a et amout of heat from the sigle reseroir at L ad coert it comletely ito work. W L L et W W L L But this is imossible; hece the corollary must be true. he hermodyamics emerature Scale Zeroth law roides a basis for temerature measuremet, but it has some short comigs, sice the measuremet of temerature deeds o the thermometric roerty of a articulars substace ad o the mode of workig of the thermometer. We kow that the of a reersible egie oeratig betwee two thermal reseroirs at differet temeratures deeds oly o the temeratures of the reseroir ad is ideedet of the ature of the workig fluid. With this ricile lord Keli deiced a temerature scale that is ideedet of the thermometric roerty of the workig substace ad this is the Keli temerature scale or thermodyamic temerature scale or absolute temerature scale. he cocet of this temerature scale may be deeloed as follows. L th L L,...() Where, absolute temerature Deartmet of Mechaical Egieerig, Page 54

55 BASIC EMODYNAMICS 0ME here are may fuctioal relatios ossible to relate L & to L &, which will sere to defie the absolute scale. the relatio that has bee selected for the thermodyamic scale of temerature is L he Carot efficiecy may be exressed as th his meas that if L th L L...()...() of a Carot cycle oeratig betwee two gie costat temerature reseroirs is kow, the ratio of the to absolute temerature is also kow, i order to assig alues of absolute temerature, howeer oe other relatio betwee L ad must be kow. Problems (o secod Law of thermodyamics) Problem. A egieer claims to hae deeloed a egie which deelos.4 kw while cosumig 0.44 Kg of fuel of calorific alue of calorific alue of 4870 kj / kg i oe hour. he maximum ad miimum temeratures recorded i the cycle are 400 C & 50 C resectiely is the claim of the egieer geuie (Set./Oct. 996) Solutio: emerature of source, = 400 C = 67 K emerature of sik, L = 50 C = 67 K We kow that the thermal efficiecy of the CANO cycle is the maximum betwee the secified temerature limits ad is gie as. carot L L L 6 i.e., carot = 6.8% he thermal efficiecy of the egie deeloed by the egieer is gie as thermal.. L W.. Deartmet of Mechaical Egieerig, Page 55

56 BASIC EMODYNAMICS 0ME We hae,. W. 4 kw x kw.4 & thermal or 66.4% 5. Sice thermal Carot, Egieer claim is ot geuie Aswer Problem. wo Carot egies A ad B are coected i series betwee two thermal reseroirs maitaied at 00 k ad 00 k resectiely. Egie A receies 680 kj of heat from the high temerature reseroir ad rejects heat to Carot egie B. Egie B takes i the heat rejected by egie A ad rejects heat to the low temerature reseroir. If egies A ad B hae equal thermal efficiecies determie () he heat rejected by egie B () he temerature at which heat is rejected by egie A ad () he work doe durig the rocess by egies A ad B resectiely. Solutio: L = B igh tem reseroir = 000 K E A E B A = 680 KJ We hae, Low temerature reseroir L = 00 K A = 680 KJ LA = B LB W A = A - LA W B = B - LB & A A B B LA A LB LA B L B i) Gie A B Deartmet of Mechaical Egieerig, Page 56

57 BASIC EMODYNAMICS 0ME LA L B i e, L LA L LA. LA B LA i.e., LA ii) We hae also We hae, L = K B [ LA = B = 6. K, i.e., the temerature at which heat is rejected by egie A] A B LA = or 68.4% he heat rejected by egie B B A i. e, i.e., LA B LA A LB B LA L L B LA A LA LA LB L LA 680 B L LA = 5.7 KJ = B Substitutig this i () we get LB KJ As. () iii) Work doe Deartmet of Mechaical Egieerig, Page 57

58 BASIC EMODYNAMICS 0ME W A = A LA = = 48.7 KJ W B = B LB = = 6.7 KJ Problem. A reersible refrigerator oerates betwee 5 C ad - C. If the heat rejected to 5 C reseroir is. kw, determie the rate at which to heat is leakig ito the refrigerator. Solutios: eersible refrigerator = 5 C = 08 K L = - C = 6 K (COP) ef = = 08 K ef L = 6 K... W. L. L. W. L. W.. i.e,. L i.e,. L i.e., L. L L.. L L ( It is a reersible refrigerator) i.e., L L = L L. L L =.0 kw As Problem 4. A reersible ower cycle is used to drie heat um cycle. he ower cycle takes i heat uits at K ad rejects at K. he heat um abstract 4 from the sik at 4 k ad discharges uits of heat to a reseroir at K. Deelo a exressio for the ratio 4 / i terms of the four temeratures. L Deartmet of Mechaical Egieerig, Page 58

59 BASIC EMODYNAMICS 0ME Solutio:- We hae For reersible ower cycle, thermal For reersible heat um cycle, (COP).P () Multily () by (), we get Cosiderig LS, () Sice Source W, we get x E Sik K 4 K W W 4 Source P Sik K K W Deartmet of Mechaical Egieerig, Page 59

60 BASIC EMODYNAMICS 0ME Deartmet of Mechaical Egieerig, Page 60 4 O substitutio i the aboe equatio 4 = 4 x i.e., As

61 BASIC EMODYNAMICS PA B UNI 5 0ME ENOPY he first law of thermodyamics itroduces the cocet of the iteral eergy U, ad this term hels us to uderstad the ature of eergy, as defied by the first law. I the similar way the secod law itroduces the cocet of etroy S, like iteral eergy it is also a thermodyamic roerty ad is defied oly i terms of mathematical oeratios. CLAUSIUS EOEM: he thermal efficiecy of reersible Carot cycle is gie by the exressio, Carot L L or L L Where ad L are the temeratures of high temerature thermal reseroir ad low temerature thermal reseroir resectiely, ad is the heat sulied ad L is the heat rejected by the Carot egie. Cosiderig the usual sig coetio, +e for the heat absorbed ad e for the heat rejected, we may write, L L or Equatio () shows that the sum of the quatities L L 0 () () ad L L, associated with absortio ad rejectio of heat by the fluid of a reersible heat egie is zero for the etire cycle. Sice the workig fluid returs to its iitial state at the ed of the cycle, it udergoes o et chage i roerties, L suggestig that the quatities ad reresets roerty chages of the workig fluid because L their sum is zero for the cycle ad this is the characteristic of a roerty or state fuctio. he amout of heat trasfer is kow to deed o the ath of the rocess. oweer if the heat is diided by the temerature at which the trasfer takes lace, the result is ideedet o the ath. he aboe coclusio is for the Carot reersible cycle. But it ca be roed that the coclusio is alid for ay reersible cycle. Cosider ay arbitrary reersible cycle a-b-c-d-a as show. I such cycle absortio ad rejectio of heat do ot occur at two costat temeratures but take lace at cotiuously chagig temeratures. Deartmet of Mechaical Egieerig, Page 6

62 BASIC EMODYNAMICS 0ME he cycle ca be ow broke ito a ifiite umber elemetary Carot cycle by drawig a series of ifiitely close adiabatic lies, eh, fg, m, etc. efgh, fmg etc reresets elemetary Carot cycle i which sectios ef, gh etc ca be cosidered as isothermal lies. For ay differetial Carot cycle, efgh, let be the heat absorbed durig isothermal rocess ef ad L be the heat rejected durig the isothermal rocess gh. he temerature of ef is ad gh rocess is L.he we may write, L L Usig roer sig coetio +e for the absortio of heat ad -e for rejectio, we get, Similarly, L L L L 0 0 () () for the cycle fmg From these relatios we see that the algebraic sum of the ratios of the amouts of heat trasferred to the absolute temerature for the Carot cycles take together is equal to zero, thus, j b m f Pressure e a L L d h olume L L g k c 0 L i.e. 0 () L Deartmet of Mechaical Egieerig, Page 6

63 BASIC EMODYNAMICS 0ME As the umber of Carot cycles is ery large, the sum of the terms oer the comlete cycle becomes equal to the cyclic itegral of, We may, therefore write 0 (4) Where reresets reersible cycle. his result is kow as CLAUSIUS EOEM. is kow as ENOPY. ENOPY: Defiitio: Etroy, S is a roerty of system such that its icrease S - S as the system chages from state to state is gie by, S S () I differetial form equatio () ca be writte as E CLAUSIUS INEUALIY: Whe ay system udergoes a cyclic rocess, the itegral aroud the cycle of equal to zero. I symbols, 0 () ds is less tha or Where δ is a ifiitesimal heat trasfer, is absolute temerature of the art of the system to which heat trasfer δ occurs. POOF: For ay reersible cycle from Clausius theorem, 0 () From the Carot s theorem we kow that the efficiecy of a irreersible egie is less tha that of a reersible egie, i.e. I Deartmet of Mechaical Egieerig, Page 6

64 BASIC EMODYNAMICS 0ME Where I is efficiecy of the irreersible egie ad is efficiecy of the reersible egie. L L ece, () I Where I ad reresets irreersible ad reersible rocesses resectiely. For a reersible egie, the ratio of the heat absorbed ad heat rejected is equal to the ratio of the absolute temeratures. herefore or L L I I L L (4) i.e. L i.e. 0 (5) I L I Usig sig coetios of +e for absortio of heat ad e for the rejectio of heat, we get, I L L I 0 (6) From this we see that the algebraic sum of the ratios of the amouts of heat trasferred to the absolute temerature for a cyclic irreersible rocess is always less tha zero, I 0 (7) Combiig equatios () ad (7), we get 0 (8) his is kow as CLAUSIUS INEUALIY. 0 If 0, the cycle is reersible, 0 L I, the cycle is irreersible ad ossible ad, the cycle is imossible sice it iolates the secod law of thermodyamics. L Deartmet of Mechaical Egieerig, Page 64

65 BASIC EMODYNAMICS 0ME ILLUSAION OF CLAUSIUS INEUALIY: Examle. Cosider the flow of heat from the reseroir at tem to that at across the coductor as show. Coductor is the system. I the steady state there is o chage i the state of the system. Let δ = 000 kj, = 500 K, = 50 K Sice δ = 000 kj, δ = -000 kj A Clausius δ δ iequality kj / K, ece 0 Examle. E is the system which executes a cyclic rocess. 500 K, roed. E 50 K 000 kj 600 kj ENOPY IS A POPEY: δw kj / K oweer, if E were a reersible egie, the work δw would hae bee, 000(500 50) W 500kJ ad 0 kj / K Deartmet of Mechaical Egieerig, Page 65

66 BASIC EMODYNAMICS 0ME Proof that etroy is a roerty: Statemet: For ay system udergoig iterally reersible cycle, the itegral of is zero, i symbols, 0 () Let the system executes a cyclic rocess, startig at state, roceedig to state alog the reersible ath A, ad returig state alog a differet ath B. Y X From the Clausius iequality we hae alog ath AB, 0 () Sice the rocess is reersible, we may reerse it ad thus cause the system to retrace its ath recisely. Let the elemet of heat trasfer corresodig to the system boudary at temerature be δ, for this reersed rocess. ' he we hae alog ath BA, 0 () But, sice the secod cycle is simle the first oe with the directio reersed, we hae, ' eersible Path A (4) eersible Path B herefore, statemet () becomes; alog ath BA, 0 (5) or 0 (6) Comarig statemets () ad (6) we see that they ca be both true simultaeously oly Deartmet of Mechaical Egieerig, Page 66

67 BASIC EMODYNAMICS 0ME if, 0 (7), hece Etroy is a roerty. Statemet: he itegral of ed states, is ideedet of the ath of the rocess. I symbols, for arbitrary aths A ad B, Y, whe a system executes ay reersible rocess betwee fixed A B Cosider a system which executes a reersible cyclic rocess, from alog ath A to, ad back alog ath C to.he we hae, AC A C 0 () Similarly, for the reersible cyclic rocess BC, we ca write, BC B C From equatios () ad () we get, i.e. () () gies, eersible Path A 0 () A C B C i.e. () A B X eersible Path B eersible Path C 0 () Path A ad ath B are arbitrary ad has the same alue for ay reersible Deartmet of Mechaical Egieerig, Page 67

68 BASIC EMODYNAMICS 0ME ath betwee () ad (), hece from the defiitio of etroy we may write ( S - S ) has the same alue for ay reersible ath betwee ad. herefore ENOPY is a roerty. CALCULAION OF ENOPY CANGE FO DIFFEEN POCESS Etroy chage i IEESIBLE rocess: For a rocess that occurs irreersibly, the chage i etroy is greater tha the heat chage diided by the absolute temerature. I symbols, ds Proof: P Irreersible Path A eersible Path B Cosider a arbitrary irreersible cycle -A--B- as show i figure. he ath to (-A-) is traersed irreersibly ad the ath to ( -B-) reersibly. From the Clausius Iequality, we hae 0, for the cycle which is irreersible ad 0, for the cycle which is reersible () Sice the etroy is a thermodyamic roerty, we ca write ds ds I ds A B 0 () For a reersible rocess we hae, ds () Substitutig this i equatio (), we get B B ds I A B 0 (4) Usig equatio (), for a irreersible cycle, Deartmet of Mechaical Egieerig, Page 68

69 BASIC EMODYNAMICS 0ME A B I 0 (5) Now subtractig equatio (5) from equatio (4), we get ds I A A I (6) For small chages i states the aboe exressio ca be writte as, ds I I (7) Where the subscrit I reresets the irreersible rocess. he equatio (7) states that i a irreersible rocess the chage i etroy is greater tha. herefore we ca write, ds I,where equality sig is for reersible rocess ad iequality sig is for irreersible rocess. MAEMAICAL EXPESSION OF E SECOND LAW: ds for reersible rocesses ad ds for irreersible rocesses he aboe equatio may be regarded as the aalytical exressio of the secod law of thermodyamics. ENOPY CANGES FO AN OPEN SYSEM: I a oe system the etroy is icreased because the mass that crosses the boudary of the system has etroy. hus for a oe system, we may write, ds Where, As the mass mass m s mi ad me i i m e s e () are the masses eterig ad leaig the system & s i, s e are the etroies. mi eters the system, the etroy is icreased by a amout m i me leaes the system, the etroy decreases by a amout m e se I steady flow rocess there is o chage i the mass of the system ad write, therefore m i is m, similarly as the e m. We ca m( s e s i ) Deartmet of Mechaical Egieerig, Page 69 ()

70 BASIC EMODYNAMICS 0ME For steady flow adiabatic rocess s e si ( sice 0 ) i.e. i a steady flow adiabatic rocess the etroy of the fluid leaig must be equal to or greater tha the etroy of the fluid comig i. Sice the equality sig holds for a reersible rocess, we coclude that for a reersible steady flow adiabatic rocess, s e s i () IMPOAN ELAIONS FO A PUE SUBSANCE INOLING ENOPY PUE SUBSANCE:A ure substace has a homogeeous ad iariable chemical comositio ee though there occurs a hase chage. he first law for a closed system is gie by, de W () I the absece of chages i kietic ad otetial eergies the equatio () ca be writte as du W For a reersible rocess herefore equatio () becomes ds du W ds () () he work doe at the boudary of a system durig a reersible rocess is gie by W Pd Substitutig this i equatio (), we get ds du Pd (4) (5) From equatio (5) we realize that it ioles oly chages i roerties ad ioles o ath fuctios. herefore we coclude that this equatio is alid for all rocesses, both reersible ad irreersible ad that it alies to the substace udergoig a chage of state as the result of flow across the boudary of the oe system as well as to the substace comrises a closed system. I terms of er uit mass the equatio (5) ca be writte as ds du Pd (6) Deartmet of Mechaical Egieerig, Page 70

71 BASIC EMODYNAMICS 0ME Sice du d d P Pd dp (sice = U + P) herefore i.e. ds ds I terms of uit mass, d d ds Pd dp dh dp dp dh dp Pd dp d or ds (7) PINCIPLE OF E INCEASE OF ENOPY: Etroy Chage for the System + Surroudigs Cosider the rocess show. Let δ is the heat trasfer from a system at temerature to the surroudigs at temerature 0, ad δw is the work of this rocess (either +e or e ). Usig the ricile of icrease i etroy System emerature = δ δw ds system ds surroudi gs for a reersible rocess Surroudigs emerature = 0 he total chage of etroy for the combied system 0 () () Deartmet of Mechaical Egieerig, Page 7

72 BASIC EMODYNAMICS 0ME ds system or ds system ds surroudigs ds surroudigs he same coclusio ca be had for a oe system, because the chage i the etroy of the system would be ds oe system m s he chage i the etroy of the surroudigs would be, ds surroudi gs herefore, ds system or ds 0 system ds i i m s ds i i surroudigs m m surroudigs e e s s e e 0 that 0 ad therefore ds system ds surroudi gs sice 0 ad it follows his meas that rocesses iolig a iteractio of a system ad its surroudigs will take lace oly if the et etroy chage is greater tha zero or i the limit remais costat. he etroy attais its maximum alue whe the system reaches a stable equilibrium state from a o equilibrium state. Problems o etroy: Problem No.. Oe kg of water at 7 K is brought ito cotact with a heat reseroir at 7 K. Whe the water has reached 7 K, fid the etroy chage of water, the heat reseroir ad of the uierse. Solutio: Let be the temerature of water, be the temerature of heat reseroir. Sice reseroir is at higher temerature tha that of water, whe water is brought ito cotact with the reseroir heat trasfer occurs from reseroir to water ad takes lace through a fiite temerature differece (irreersible). he etroy of water would icrease ad that of reseroir decrease so that et etroy chage of the water ad the reseroir together would be +e defiite. o fid the etroy chage of water we hae to assume a reersible ath betwee ed states, which are at equilibrium. 0 Deartmet of Mechaical Egieerig, Page 7

73 BASIC EMODYNAMICS 0ME eat reseroir = 7 K ) Etroy of water System (water) = 7 K ) he temerature of the reseroir remais same irresectie of the amout of the heat withdraw. Amout of heat absorbed by the system from the reseroir, mc x 4, kJ herefore, Etroy chage of reseroir, 48.7 S reseroir.5kj / K (-e sig idicates decrease i etroy) 7 ) Etroy chage of the uierse Suierse Swater Sreseroir kJ / K Problem No.. wo kg of air is heated from 7 0 C to 47 0 C while the ressure chages from 00 kpa to 600 kpa. Calculate the chage of etroy. = 0.57 kj / kg K, C =.005 kj / kg K. Solutio: Gie: m = kg = = 00K = = 700K S water mc 7 x 4.87l P = 00 kpa P = 600 kpa d mc l.068kj / K he geeral equatio used for the calculatio of chage of etroy is gie by, Deartmet of Mechaical Egieerig, Page 7

74 BASIC EMODYNAMICS 0ME S air S S mc 0.78kJ / K l P m l P x.005l x 0.57l Problem No.. e grams of water at 0 0 C is coerted ito ice at C at costat atmosheric ressure. Assumig secific heat of liquid water to remai costat at 4.84 J / g 0 C ad that of ice to be half of this alue, ad takig the latet heat of fusio of ice at 0 0 C to be 5 J / g, calculate the total etroy chage of the system. Solutio: Gie : m = 0 gm Water is aailable at temerature = 0 0 C = 9 k C (water) = 4.84 J / g 0 C = 4.84 kj / kg 0 C Ice is to be formed at temerature 4 = -0 0 C = 6 K C (ice) = ½ x 4.84 J / g 0 C = ½ x 4.84 kj / kg 0 C otal etroy chage of water (system) as it is coerted ito ice will be S otal S I S II S III S II S I S III a) SI i.e. etroy chage of the system as it is cooled from 0 0 C to 0 0 C. () Deartmet of Mechaical Egieerig, Page 74

75 BASIC EMODYNAMICS 0ME S I 7 9 mc d.958x0 mc kj / K 7 l x 4.84l 7 9 S b) II i.e. etroy chage of water at 0 0 C.to become ice at 0 0 C ml 0 5 SII x 0.0kJ / K S c) III S III 6 7 i.e. etroy chage of ice as it is cooled from 0 0 C to -0 0 C mc ( ice ) 7.807x0 d 4 mc kj / K ( ice ) 6 l x l herefore total etroy chage of water as it is coerted ito ice will be S otal S I S II S kJ / K III.958x0 ( 0.0) 7.807x0 Problem No.4. A reersible egie as show i figure durig a cycle of oeratio draws 5MJ from the 400 K reseroir ad does 840 kj of work. Fid the amout ad directio of heat iteractio with other reseroirs. 00 K 00 K 400 K = 5 MJ E W = 840 kj ( 4 ) Solutio: Let us assume that ad are the heat rejected by the egie to the reseroir at 00 K ad 00 K resectiely. Deartmet of Mechaical Egieerig, Page 75

76 BASIC EMODYNAMICS 0ME From the Clausius theorem we hae 0 i.e. 0 () Ad also, W () Cosider equatio (), MJ, 400K, 00K, 00K 5 Usig e sig for heat rejected i the equatio, we hae i. e.500 5x , x ad equatio () gies, 5x (4) Solig equatios () ad (4), we get x.8mj ad 4. 98MJ 0 herefore the directios of heat iteractio with the reseroirs are as follows 00 K = 0.8 MJ 00 K = 4.98 MJ E () = 5 MJ 400 K W = 840 kj Deartmet of Mechaical Egieerig, Page 76

77 BASIC EMODYNAMICS UNI 6 he Pure Substace 0ME he system ecoutered i thermodyamics is ofte quite less comlex ad cosists of fluids that do ot chage chemically, or exhibit sigificat electrical, magetic or caillary effects. hese relatiely simle systems are gie the geeric ame the Pure Substace. Defiitio A system is set to be a ure substace if it is (i) homogeeous i chemical comositio, (ii) homogeeous i chemical aggregatio ad (iii) iariable i chemical aggregatio. omogeeous i chemical comositio meas that the comositio of each art of the system is same as the comositio of ay other art. omogeeous i chemical aggregatio imlies that the chemical elemets must be chemically combied i the same way i all arts of the system. Iariable i chemical aggregatio meas that the chemical aggregatio should ot ary with resect to time. (i) (ii) (iii) Satisfies coditio (i) Satisfies coditio (i) Does ot satisfies coditio (i) Satisfies coditio (ii) Satisfies coditio (iii) Steam + ½ O (Gas) Does ot satisfies coditio (ii) + O (Gas) Figure Illustratio of the defiitio of ure substace I figure three systems are show. he system (i) show i the figure is a mixture of steam ad water. It is homogeeous i chemical comositio because i eery art of the system we hae, for eery atom of oxyge we hae two atoms of hydroge, whether the samle is take from steam or water. he same is through for system (ii) cosistig of water ad ucombied mixture of hydroge ad oxyge. System (iii) howeer is ot homogeeous i chemical comositio because i the uer art of the system hydroge ad oxyge are reset i the ratio : where as i the bottom ortio they are reset i the ratio :. System (i) also satisfies coditio (ii), because both hydroge ad oxyge hae combied chemically i eery art of the system. System (ii) o the other had does ot satisfies coditio (ii) because the bottom art of the system has two elemets amely hydroge ad oxyge hae chemically combied where as i the uer art of the system the (ii) elemets aear as a mixture of two idiidual gases. Deartmet of Mechaical Egieerig, Page 77

78 BASIC EMODYNAMICS 0ME Iariable i chemical aggregatio meas that the state of chemical combiatio of the system should ot chage with time. hus the mixture of hydroge ad oxyge, if it is chagig ito steam durig the time the system was uder cosideratio, the the systems chemical aggregatio is aryig with time ad hece this system is ot a ure substace. hus the system (i) is a ure substace where as the systems (ii) ad (iii) are ot ure substaces. Secific heat, C Whe iteractio of heat takes lace betwee a closed system ad its surroudigs, the iteral eergy of the system chages. If δ is the amout of heat trasferred to raise the temerature of kg of substace by d, the, secific heat C = δ/d As we kow, the secific heat of gas deeds ot oly o the temerature but also uo the tye of the heatig rocess. i.e., secific heat of a gas deeds o whether the gas is heated uder costat olume or uder costat ressure rocess. We hae d = m C. d ad d = m C. d for a re. o-flow rocess at costat olume for a re. o-flow rocess at costat ressure For a erfect gas, C & C are costat for ay oe gas at all ressure ad temeratures. ece, itegratig aboe equatios. Flow of heat i a re. costat ressure rocess = m C ( ) Flow of heat i a re. costat olume rocess = m C ( ) he iteral eergy of a erfect gas is a fuctio of temerature oly. i.e, u = f (), to ealuate this fuctio, let kg of gas be heated at costat olume From o-flow eergy equatio, δ = du + δw δw = 0 sice olume remais costat δ = du = C. d It. U = C + k where k is a costat For mass m, It. eergy = m C Ay rocess betwee state to state, Chage i it. eergy = m C ( ) (U U ) = m C ( ) We ca also fid the relatioshi betwee C & C & show that C C C = ; ; C C r & C P r r Deartmet of Mechaical Egieerig, Page 78

79 BASIC EMODYNAMICS 0ME Ethaly: Cosider a system udergoig a quasi equilibrium costat ressure rocess. We hae from st law of thermodyamics for a o-flow rocess, - = U U + W - W - = d Sice ressure is costat W - = ( ) - = U U + ( ) = (U + ) (U + ) i.e., heat trasfer durig the rocess is gie i terms of the chage i the quatity (U + ) betwee iitial ad fial states. herefore, it fid more coeiet i thermodyamics to defie this sum as a roerty called Ethaly () i.e., = U + I a costat ressure quasi equilibrium rocess, the heat trasfer is equal to the chage i ethaly which icludes both the chage i iteral eergy ad the work for this articular rocess. he ethaly of a fluid is the roerty of the fluid, sice it cosists of the sum of a roerty ad the roduct of the two roerties. Sice ethaly is a roerty, like iteral eergy, ressure, secific olume ad temerature, it ca be itroduced ito ay roblem whether the rocess is a flow or a oflow rocess. For a erfect gas, we hae h For ay rocess, δ = d = u + = C + = (C + ) = C i.e., h = C & = mc = mc d For a rocess betwee states & Chage i ethaly = ( ) = mc ( ) Secific heat at Costat olume: Whe heat iteractio takes lace at costat olume, δw = 0 ad from st law of thermodyamics, for uit mass, (δq) = du he amout of heat sulied or remoed er degree chage i temerature, whe the system is ket uder costat olume, is called as the secific heat at costat olume, Deartmet of Mechaical Egieerig, Page 79

80 BASIC EMODYNAMICS 0ME Or C = d du d Or du = C d Secific heat at Costat ressure Whe heat iteractio is at costat ressure, (δq) = dh he amout of heat added or remoed er degree chage i temerature, whe the system is ket uder costat ressure, is called as the secific heat at costat ressure. Or C = d Or dh = C. d dh d Alicatio of st law of thermodyamics to o-flow or closed system: a) Costat olume rocess ( = costat) Alyig st law of thermodyamics to the rocess, - = U U + W - = U U + 0 i.e., - = C ( ) For mass m of a substace, = mc ( ) b) Costat ressure ( = Costat) Alyig st law of thermodyamics to the rocess, - = u u + W - he work doe, W - = d = ( ) i.e., - = u u + ( ) = (u + ) (u + ) = h h i.e., = C ( ) For mass m of a substace, = mc ( ) c) Costat temerature rocess (Isothermal rocess, = costat) Alyig st law of thermodyamics to the rocess, - = U U + W - Deartmet of Mechaical Egieerig, Page 80

81 BASIC EMODYNAMICS 0ME = C ( ) + W - i.e., - = W - - = l / = l / d) eersible adiabatic rocess ( ) = costat Alyig st law of thermodyamics to the rocess, - = U U + W - O = u U + W () Or (U U ) = (U U ) = he aboe equatio is true for a adiabatic rocess whether the rocess is reersible or ot. I a adiabatic exerimet, the work doe W - by the fluid is at the exese of a reductio i the iteral eergy of the fluid. Similarly i a adiabatic comositio rocess, all the work doe o the fluid goes to icrease the iteral eergy of the fluid. o derie = C: For a reersible adiabatic rocess We hae δq = du + δu For a reersible rocess, δw = d δq = du+ d = O For a adiabatic rocess δq = 0 Also for a erfect gas, = or = d du + Also, u = C or du = C d d C d + d d or C 0 Deartmet of Mechaical Egieerig, Page 8

82 BASIC EMODYNAMICS 0ME It., C l + l = costat Sub. = / C l Or l P l = costat C l = costat Also, C = or C l l or l i.e., l or x ( )l = costat l = costat = costat r = costat r = e costat = costat i.e., = costat we hae = or = sub. his alue of i = C = C or - = costat --- (a) Also, = sub. his i equatio ressure = C P = costat Deartmet of Mechaical Egieerig, Page 8

83 BASIC EMODYNAMICS 0ME = costat or = costat --- (b) For a reersible adiabatic rocess for a erfect gas betwee states &, we ca write = - = - r r r r or or or r r r --- (d) --- (e) he work doe i a adiabatic rocess is W = u u he gai i I.E. of a erfect gas, is u u = C ( ) But C = W = Usig =, W = ( ) W = C ( ) --- (c) e) Poly troic rocess ( = costat) Alyig st law of thermodyamics, - = u u + W - = (u u ) + i.e., = Also C = - C ( ) sub. & simlifyig = W I a oly troic rocess, the idex deeds o the heat ad work quatities durig the rocess. Deartmet of Mechaical Egieerig, Page 8

84 BASIC EMODYNAMICS 0ME. A cylider cotais 0.45 m of a gas at bar & 80 0 C. he gas is comressed to a olume of 0. m, the fial ressure beig 5 bar. Determie i) the mass of the gas, ii) the alue of idex for comositio, iii) the icrease i iteral eergy of the gas ad i) the heat receied or rejected by the gas durig comressio. (ake =.4, = 94. J/kg-K). Solutio: = 0.45 m = x 0 5 Pa = 0. m = 5 K = 5 x 0 5 Pa 5 x0 x0.45 i) We hae = m m = = 0.4 kg 94.x5 ii) iii) = Or i.e., =.96 Icrease i it. eergy, U = mc ( ) i) We hae = U + W W = = kj = = kj eat rejected = kj = 0.4 x = K ( ) r 94. = 0.4 x = 49.9 kj = m = he roerties of a certai fluid are related as follows U = t = 0.87 (t + 7) where u is the s. Iteral eergy (kj/kg), t is i 0 C, is ressure (kn/m ) ad is s. olume (m /kg). For this fluid, fid C & C Solutio: By defiitio s. eat at costat olume C = du dt du dt Deartmet of Mechaical Egieerig, Page 84

85 BASIC EMODYNAMICS 0ME d C = ( t) dt = 0.78 kj/kg 0 C Also, C = dh dt d dt du d = dt dt u d d = t 0.87t 0.87x7 dt dt d 0.78t dt =.005 kj/kg 0 C 0.87t. A fluid system cosistig of 4.7 kg of a ure substace has a eergy E of 85 kj. he kietic eergy of the system is 7 kj ad its graitatioal otetial eergy is 5 kj. he system udergoes a adiabatic rocess i which the fial s. i.e., is 50 kj/kg, the fial kietic eergy is.9 kj ad the fial graitatioal otetial eergy is. kj. he effects due to electricity, caillary ad magetism are assumed to be abset. a) Ealuate the iitial alue of the s. i.e., of the fluid. b) Determie the magitude ad sig of the work doe durig the rocess. Solutio: otal iitial eergy E =KE + PE +U Iitial s. i.e., = 85 = U U = 6 kj kJ / kg Fial state: E = kε + Pε + U = (50) = 68.5 kj From st law, δ = E E + W 0 = W = kj A mass of 0. kg of a ure substace at a ressure of bar ad a temerature of k occuies a olume of 0.5 m. Gie that the it. eergy of the substace is.5 kj, ealuate the s. Ethaly of the substace. Solutio: m = 0. kg P = x 0 5 N/m = k = 0.5 =.5 kj We hae, ethaly = U + P =.5 x 0 + x 0 5 x 0.5 = 46.5 kj s. Ethaly = 46.5/0. =.5 kj/kg Deartmet of Mechaical Egieerig, Page 85

86 BASIC EMODYNAMICS 0ME 4. A gas eters a system at a iitial ressure of 0.45 MPa ad flow rate of 0.5 m /s ad leaes at a ressure of 0.9 MPa ad 0.09 m /s. Durig its assage through the system the icrease i i.e., is 0 kj/s. Fid the chage of ethaly of the medium. Solutio: = 0.45 x 0 6 Pa = 0.5 m /s = 0.9 x 0 6 Pa = 0.09 m /s (u u ) = 0 x 0 J/s We hae from st law for a costat ressure quasi static rocess - = (u + ) (u + ) = ( ) = Chage i ethaly = (u u ) + = 0 x x 0 6 x x 0 6 x 0.5 ( ) = -.5 kj/s here is a decrease i ethaly durig the rocess Deartmet of Mechaical Egieerig, Page 86

87 BASIC EMODYNAMICS 0ME UNI 7 EMODYNAMIC ELAIONS Ideal Gas Defiitio: A substace is said to be a ideal gas if it satisfies the followig equatios i.e., P = ad u = f () Where P is the ressure exerted by the substace, is the secific olume of the substace, is the temerature i degree Keli, u is the secific iteral eergy ad is the gas costat. Exeriece has show that almost all real gases satisfy the aboe equatios oer wide rages of ressures ad temeratures. oweer there are certai situatios where the real gases caot be treated as ideal gases. Mole of a Gas A mole of a gas is that quatity of gas whose mass is umerically equal to its molecular weight. For examle, kg mol of hydroge is equal to kg, has molecular weight of hydroge is. herefore if is the total umber of moles, m is mass ad M is the molecular weight the, M=m Aogadro s yothesis Aogadro s law states that equal olumes of all gases measured at the same temerature ad ressure cotai the same umber of moles. Cosider two gases A ad B. he law states that if A = B, A = B ad P A = P B the A = B For gas A, the equatio of state ca be writte as P A A = m A A A = A M A A A Or A PA A M Similarly for gas B we hae A A A B PB B M B B B Accordig to Aogadro s law, A = B, A = B ad P A = P B the A = B. herefore it follows that M A A = M B B = Where is called the uiersal gas costat ad is called the characteristic gas costat. he alue of =8.4 kj/kgmol-k. he ideal gas equatio ca also be writte i terms of as = M = where = /M Secific eat of Ideal Gases From the defiitio of secific heat at costat olume ad the secific heat at costat ressure, we hae Deartmet of Mechaical Egieerig, Page 87

88 BASIC EMODYNAMICS 0ME c c P du d dh d f f as u ad h are fuctios of temerature. From the aboe equatios, du = c d ad dh = c P d For a mass of m kg of gas the equatios become du = m c d ad d = m c P d O itegratig we get U U ad m c d m c P d elatio betwee secific heats for a ideal gas For a ideal gas h = u + herefore Or dh d c P = c + ie, c P - c = du d Diidig the aboe equatio by c, we get Or c c Similarly, diidig the aboe equatio by c P, we get c P Chages i iteral eergy, ethaly ad etroy for a ideal gas i) Chage i iteral eergy Deartmet of Mechaical Egieerig, Page 88

89 BASIC EMODYNAMICS 0ME Let a ideal gas of fixed mass m udergoes a fiite chage of state from temerature to temerature. he the chage i iteral eergy is gie by U U m u u m du m c d o itegrate the aboe equatio we should kow the fuctioal relatioshi betwee c ad d. A erfect gas equatio is oe for which c is a costat. herefore, U -U = m c ( ) ii) Chage i Ethaly We hae m m h h m dh c P d For a erfect gas c P is costat. herefore the aboe equatio ca be itegrated ad we get - = m c P ( ) Work doe by a erfect gas durig a reersible adiabatic rocess i a closed system: From the first law of D, q du w For a adiabatic rocess, δq=0 ece the work doe by a uit mass of a erfect gas o a isto durig a adiabatic exasio rocess is equal to the decrease i iteral eergy, i.e., δw = -du Whereas, for a adiabatic comressio rocess, the iteral eergy of the gas will icrease with a cosequet icrease i temerature. For a erfect gas, du=c d w c d But, c w d Deartmet of Mechaical Egieerig, Page 89

90 BASIC EMODYNAMICS 0ME ece the work doe durig a adiabatic o-flow rocess betwee states ad is gie by W P P P P P P P P Work doe by a erfect gas durig a reersible adiabatic Steady Flow rocess: Neglectig the effect of chages i elocity ad eleatio, SFEE for a uit mass of fluid is gie by w h q Sice the rocess is adiabatic, q=0. ece work doe er uit mass of a erfect gas durig a adiabatic steady flow exasio rocess is equal to the decrease i ethaly, i.e.,w - = h -h For a erfect gas, dh = c d herefore, w - = c ( - ) But, c, the work doe durig a adiabatic steady flow rocess betwee states ad is gie by, w P P P P P P P P Deartmet of Mechaical Egieerig, Page 90

91 BASIC EMODYNAMICS 0ME It may be oted that the work doe for a steady flow system is times that for a closed system. he olytroic rocess of a erfect gas: A Polytroic rocess is oe for which the ressure olume relatio is gie by = costat, where the exoet for the gie rocess is a costat ad may hae ay umerical alue ragig from lus ifiity to mius ifiity. From the aboe equatio, it is eidet that the roerties at the ed states of the reersible or irreersible olytroic rocess of a erfect gas may be writte i the form P P P P here are four alues of the exoet that idicate rocesses of articular iterest. Whe = 0, costat ressure or isobaric rocess = ±, costat olume or isoolumic rocess =, costat temerature or isothermal rocess ad =, costat etroy or isetroic rocess hese rocesses are show i the fig. o - ad -s diagrams. Figure: Polytroic Process o - ad -s diagrams Deartmet of Mechaical Egieerig, Page 9

92 BASIC EMODYNAMICS 0ME Work doe ad heat trasfer by a erfect gas durig a olytroic rocess: For the closed system, the work doe durig a olytroic exasio rocess is gie by, w d ( ) P P P P P he work doe for a steady flow system durig a olytroic exasio rocess is gie by, w d P P P P P It may be oted that the work doe for a steady flow system is times that for a closed system. i) he heat trasfer for a closed system: From the first law of thermodyamics for a uit mass of substace δq = δw + du Sice du = c d ad for a reersible rocess δw = d δq = c d + d herefore heat trasfer er uit mass durig a olytroic rocess i a closed system from the iitial state to fial state is gie by q c d Deartmet of Mechaical Egieerig, Page 9

93 BASIC EMODYNAMICS 0ME Deartmet of Mechaical Egieerig, Page 9 d w But c q c c c c w q Or ii) he heat trasfer i a steady flow rocess: From the first law for steady flow system for a uit mass of fluid δq = δw + dh But dh = c d ad for a reersible steady flow rocess δw = -d herefore δq = c d - d d c q d w But herefore heat trasfer, c q c c c

94 BASIC EMODYNAMICS 0ME c Chage i Etroy w Let P,,, S aly to the iitial coditios of certai amout of gas. P,,, S, aly to the fial coditios after addig some heat. From first law of D, δq = δw + du Also by defiitio, du = c d herefore δq = P.d + c d Diide by, q But ds ds P d q c d P d δq = P.d + du c d From erfect gas equatio for uit mass of gas, P =, therefore P/ = / ds d c d herefore itegratig, ds d i.e., the chage i etroy is gie by c d S S l c l --- () Equatio () ca also be exress i terms of ressure ad olume. We hae P P Deartmet of Mechaical Egieerig, Page 94

95 BASIC EMODYNAMICS 0ME P P Substitutig this i equatio () we get, i. e., S S c l c l But c P c =, i.e., c + = c P S S c l c l P P P P --- () P Equatio () ca also be rereseted i terms of temerature ad olume. We hae P P P P Substitutig this i equatio () we get S P S.l c l P Or S S cp l. l --- () P P Ealuatio of chage i etroy i arious quasi static rocesses. Costat olume Process We hae δ = δw + du δw = 0 for a costat olume rocess. herefore heat added, δ = du = m c d Diide by mc d Itegratig, mc d Deartmet of Mechaical Egieerig, Page 95

96 BASIC EMODYNAMICS 0ME S S mc. l. Costat Pressure Process δ = δw + du = P.d + du = m d + mc d = m ( + c ) d = m (c P c + c )d = mc P d Diide by, Itegratig, S S mcp. l mc P mc d d P. Isothermal Process We hae δ = δw + du But du = 0 P Diide by, l Itegratig we get, m l S S m l 4. eersible Adiabatic Process δ = 0 herefore S S = 0 ece the rocess is called isetroic rocess 5. Polytroic Process We kow that for a erfect gas P. d Deartmet of Mechaical Egieerig, Page 96

97 BASIC EMODYNAMICS 0ME Diide by, P. d d P We kow P =, i.e., P = / S S S d d S. l Show that the etroy chage for a ideal gas udergoig a olytroic rocess accordig to the equatio P = c is gie by S S. l We hae the chage i etroy for uit mass of a substace betwee states () ad () is gie by S S ds For a erfect gas, we kow that Diide by ad itegratig, S S P. d P. d. d P P P. d W Deartmet of Mechaical Egieerig, Page 97

98 BASIC EMODYNAMICS 0ME Semi-erfect gas: S d d S l It ca be obsered that from the defiitio of c ad c that the secific heats ca be either costats or fuctios of temerature. A semi-erfect gas is oe which follows the ideal gas relatio with its secific heats beig fuctios of temerature. i.e., P = Ad c = () c = f() For examle the costat ressure molal secific heat of air at low ressure is related to the temerature by the emirical relatio Where c x0 0.9x0 c is i kj/kg-mole K ad is i Keli. It ca be see that itegratio of secific heat equatio is time cosumig. ece Keea ad Kaye deeloed the gas tables (able C-) to take ito accout the ariatio of secific heats with temerature. Iteral eergy ad ethaly of arious gases icludig air at low ressure for wide rage of temerature are tabulated i these tables. he table illustrates the roerties of air takig ito accout the ariatio of secific heats with temerature. Gas table may also be used for isetroic rocesses of erfect gases to relate roerties by itroducig a relatie ressure r ad a relatie olume r. he etroy at a referece state where the temerature is 0 ad the ressure is bar is assumed as zero. herefore at a temerature ad ressure, the etroy s is gie by s 0 c d l I gas tables a etroy fuctio s 0 is defied as Deartmet of Mechaical Egieerig, Page 98

99 BASIC EMODYNAMICS 0ME Deartmet of Mechaical Egieerig, Page 99 d c s 0 0 Sice c is a fuctio of temerature, s 0 is also a fuctio of temerature for a erfect gas. From the aboe equatios we get s = s 0 l herefore the chage etroy of a erfect gas betwee states () ad () ca be writte as 0 0 l s s s s For a isetroic rocess, s = 0 herefore ) ( l 0 0 f s s s A relatie ressure r is defied a ratio of the ressure to the referece ressure 0. 0.,. e i r herefore from the aboe equatio we ca write, ) ( l / / l l 0 0 f r r r r s Further from the equatio state of erfect gas, we hae r r he relatie olume r is defied as r r Usig this equatio i the aboe equatio we get ) ( f r r s Problems

100 BASIC EMODYNAMICS 0ME..5 m of air at 80 0 C at 8 bar is udergoig a costat ressure util the olume is doubled. Determie the chage i the etroy ad ethaly of air. Solutio: Assumig air behaes like a erfect gas we hae, C P =.005 kj/kg 0 K, C = 0.78 kj/kg 0 K ad = 0.87 kj/kg 0 K Gie: =.5 m, = = 45 0 K, P = P = 8 x 0 5 N/m ad = For a costat ressure rocess, chage i etroy is S S mc P l We hae P = m 5 8x0 x.5 m 7. 69kg 87x45 Also P P x.5x herefore chage i etroy = 0 K x.005l 45 = kj/ 0 K Chage i ethaly = - = mc P ( ) = kj. kg of air iitially at 7 0 C is heated reersibly at costat ressure util the olume is doubled, ad the is heated at costat olume util the ressure is doubled. For the total ath fid i) Work trasfer, ii) eat trasfer, iii) Chage i etroy Solutio: Gie: m = kg, = 00 0 K =, P = P = P P Process -: Costat ressure rocess i) Work doe, W - = P ( ) = P P Deartmet of Mechaical Egieerig, Page 00

101 BASIC EMODYNAMICS 0ME = m ( ) Also P P But P = P But herefore = = K herefore work doe W - = x 0.87 x (600 00) = 86. kj ii) From first law of D, eat rasfer, - = W - + (U U ) iii) Chage i etroy, S S mcp l Process -: Costat olume Process Gie, P = P, = 600 K We hae P P P P xp P But 600 (.005)l 00 = kj/ 0 K = W - + mc ( ) = x 0.78 x (600 00) = 0.5 kj = x = 00 0 K i) Work doe, W - = 0 ii) eat trasfer, - = W - + (U U ) = mc ( ) = 40.8 kj iii) Chage i etroy S S mc P l = kj/ 0 K Deartmet of Mechaical Egieerig, Page 0

102 BASIC EMODYNAMICS 0ME herefore work trasfer i total ath, W - = W - + W - = = 86. kj eat trasfer i total ath, - = = = 7. kj Chage i etroy for the total ath = (S S ) = (S S ) + (S S ) = =.94 kj/ 0 K. A mass of air is iitially at 60 0 C ad 700 kpa, ad occuies 0.08m. he air is exaded at costat ressure to 0.084m. A olytroic rocess with =.5 is the carried out, followed by a costat temerature rocess which comletes a cycle. All the rocesses are reersible. (i) sketch the cycle i the P- ad -s diagrams. (ii) fid the heat receied ad heat rejected i the cycle. (iii) fid the efficiecy of the cycle. Solutio: P = 700 kpa, = 5K =, = 0.08m, = 0.084m Now We hae P = m P P 700x0.08 m 0. 8kg 0.87x herefore = x 5 = 559 K Deartmet of Mechaical Egieerig, Page 0

103 BASIC EMODYNAMICS 0ME Agai P P () 7 eat trasfer i rocess -, - = mc ( ) eat trasfer i rocess -, mc mc O substitutig - = kj For rocess - - = du + W - U = 0.8 x.005 (599 5) = 7.kJ d m P But du = 0, i.e., W d m l m l P O substitutig, - = kj eat receied i the cycle 7. kj eat rejected i the cycle = = 84. kj he efficiecy of the cycle cycle or9% 4. kg of air at a ressure of 7 bar ad a temerature of 90 0 C udergoes a reersible olytroic rocess which may be rereseted by P. = C, fial ressure is.4 bar. Ealuate i) he fial secific olume, temerature ad icrease i etroy, ii) Work doe ad heat trasfer. Solutio: Gie, m = kg, P = 7 bar, = 6 0 K, P. = C, P =.4 bar Deartmet of Mechaical Egieerig, Page 0

104 BASIC EMODYNAMICS 0ME Air is erfect gas i.e., P = m (87)(6) 0.488m / kg 5 7x0 Also we hae, P.. = P herefore = m Also P = m herefore =.6 0 K Chage i etroy for a olytroic rocess is, S S. l Substitutig the alues, otig =.4 ad = 0.87 kj/kg 0 K, we get S S = 0.495kJ/kg 0 K Work doe, W P P Substitutig we get, W - = 4.75 kj/kg eat trasfer - = W - + (U U ) = W - + mc ( ) = = 06.9 kj/kg Deartmet of Mechaical Egieerig, Page 04

105 BASIC EMODYNAMICS 0ME UNI 8 hermodyamics of No-reactie Mixtures IDEAL GAS MIXUE Assumtios:. Each idiidual costituet of the mixture behaes like a erfect gas.. he mixture behaes like a erfect gas.. Idiidual costituets do ot react chemically whe the mixture is udergoig a rocess. Mixture characteristics: Figure: omogeeous gas mixture Cosider a mixture of gases a, b, c,. existig i equilibrium at a ressure P, temerature ad haig a olume as show i figure. he total mass of the mixture is equal to the sum of the masses of the idiidual gases, i.e., m m = m a + m b + m c +.. where subscrit m = mixture, a, b, c = idiidual gases. Mass fractio: he mass fractio of ay comoet is defied as the ratio of the mass of that comoet to the total mass of the mixture. It is deoted by m f. hus, m fa ma mb m fa, m fb, m m m fb a, b, c,. m fc P,,... m fi i m fc m c m Where the subscrit i stads for the i th comoet. It shows that the sum of the mass fractio of all comoets i a mixture is uity. Mole fractio: If the aalysis of a gas mixture is made o the basis of the umber of moles of each comoet reset, it is termed a molar aalysis. he total umber of moles for the mixture is equal to the sum of the umber of moles of the idiidual gases i.e., m = a + b + c +.. where subscrit m = mixture, a, b, c = idiidual gases. Deartmet of Mechaical Egieerig, Page 05

106 BASIC EMODYNAMICS 0ME (A mole of a substace has a mass umerically equal to the molecular weight of the substace, i.e., kg mol of O has a mass of kg, kg mol of N has a mass of 8 kg, etc.,) he mole fractio of ay comoet is defied as the ratio of the umber of moles of that comoet to the total umber of moles. It is deoted by y ad a b i. e., ya, yb, m m y a y b y c y c... yi i i.e., the sum of the mole fractio of all comoets i a mixture is uity. c m he mass of a substace m is equal to the roduct of the umber of moles ad the molecular weight (molar mass) M, or m = M For each of the comoets we ca write, m M m = a M a + b M b + c M c +... Where M m is the aerage molar mass or molecular weight of the mixture. Or M m = y a M a + y b M b + y c M c hus, the aerage molecular weight of a gas mixture is the sum of the roducts of all the comoets of the mole fractio ad corresodig molecular weight of each comoet. Note: Uiersal gas costat =8.4 kj/kg-mole K Partial Pressure: a+b Mixture P,, M where M = molecular weight, : secific gas costat, ad a & b = gases of the mixture = otal olume of the mixture = emerature of the mixture Partial ressure of a costituet i a mixture is the ressure exerted whe it aloe occuies the mixture olume at mixture temerature. If P a is artial ressure of gas a, the P a = m a a Where m a = mass of gas a, a = gas costat for gas a, similarly P b = m b b Partial olume: Partial olume of a gas i a mixture is the olume occuied by the gas comoet at mixture ressure ad temerature. Let a = artial olume of gas a ad b = artial olume of gas b i.e., P a = m a a & P b = m b b Deartmet of Mechaical Egieerig, Page 06

107 BASIC EMODYNAMICS 0ME he Gibbs-Dalto Law Cosider a mixture of gases, each comoet at the temerature of the mixture occuyig the etire olume occuied by the mixture, ad exertig oly a fractio of the total ressure as show i figure.,, = Alyig the equatio of state for this mixture we may write, P m = m m m = m M m m = m Similarly a b c c b a We kow that m = a + b + c +... ece m a, a,, m a b Or m = a + b + c +..., = i c... he aboe equatio is kow as the Gibbs Dalto Law of artial ressure, which states that the total ressure exerted by a mixture of gases is equal to the sum of the artial ressures of the idiidual comoets, if each comoet is cosidered to exist aloe at the temerature ad olume of the mixture. Gas costat for the mixture: We hae P a = m a a P b = m b b Or (P a + P b ) = (m a a + m b b ) m b, b, Also, sice the mixture behaes like a erfect gas, We hae P = m --- () m c, c, m m, m, By Dalto s law of artial ressure, which states that, the ressure of mixture of gas is equal to the sum of the artial ressures of the idiidual comoets, if each comoet is cosidered to exist aloe at the temerature ad olume of the mixture., i.e., P = P a + P b Deartmet of Mechaical Egieerig, Page 07

108 BASIC EMODYNAMICS 0ME P = (m a a + m b b ) --- () From equatio () ad (), m = m a a + m b b m a a m m Also for gas mixture, P a = m a a Similarly P a P P a y a b P a b = a M a a a M Similarly it ca be show that mole fractio = olume fractio ece, y a P Molecular weight of the mixture: a P We hae, P a = m a a a P a = a M a a Similarly P b = b M b b (P a + P b ) = ( a M a a + b M b b ) Also P = M By Dalto s law of artial ressure, P = P a + P b M = ( a M a a + b M b b ) M = y a M a a + y b M b b a M y M Also, m = m a a + m b b a a a y b M b b = m fa a + m fb b But ; a, M M a b M b Deartmet of Mechaical Egieerig, Page 08

109 BASIC EMODYNAMICS 0ME M m f. m f. a b M a M b i. e., M M m fa m M fa a M. M b a M m M b m fb fb b. M a m fa. M b M a m M b fb. M a he Amagat-Leduc Law: Exresses the law of additie olume which states that the olume of a mixture of gases is equal to the sum of the olumes of the idiidual comoets at the ressure ad temerature of the mixture. i.e., m = a + b + c. P, = For Dalto law, P m = P a + P b +P c +., = i i i P i Gibb s Law: It states that the iteral eergy, the ethaly ad the etroy of a mixture of gas is equal to sum of the iteral eergies, the ethalies ad etroies resectiely of the idiidual gases ealuated at mixture temerature ad ressure. U = U a + U b du d mu = m a U a + m b U b U = m fa U a + m fb U b m fa du d a m fb du d Similarly C P = m fa (C ) a + m fb (C ) b If b C m fa ( C a m fb ( C C Secific heat at costat olume o mole basis C Secific heat at costat ressure o mole basis ) C C P P y y a a C C P a a y y b b C C P b b & ) b Deartmet of Mechaical Egieerig, Page 09

110 BASIC EMODYNAMICS 0ME Itroductio A ideal gas is a gas haig o forces of itermolecular attractio. he gases which follow the gas laws at all rage of ressures ad temeratures are cosidered as ideal gases. A ideal gas obeys the erfect gas equatio P = ad has costat secific heat caacities. A real gas is a gas haig forces of iter molecular attractio. At ery low ressure relatie to the critical ressure or at ery high temeratures relatie to the critical temerature, real gases behae early the same way as a erfect gas. But sice at high ressure or at low temeratures the deiatio of real gases from the erfect gas relatio is areciable, these coditios must be obsered carefully, otherwise errors are likely to result from iaroriate alicatio of the erfect gas laws. Due to these facts, umerous equatios of state for real gas hae bee deeloed, the deriatio of which is either aalytical, based o the kietic theory of gases, or emirical, deried from a exerimetal data. Deartmet of Mechaical Egieerig, Page 0

111 BASIC EMODYNAMICS 0ME ader Waals Equatio of State: I deriig the equatio of state for erfect gases it is assumed that the olume occuied by the molecules of the gas i comariso to the olume occuied by the gas ad the force of attractio betwee the adjacet molecules is ery small ad hece the molecules of gas are eglected. At low ressures, where the mea free ath is large comared to the size of the molecules, these assumtios are quite reasoable. But at high ressure, where the molecules come close to each other, these are far from correct. ader waals equatio itroduces terms to take ito accout of these two modifyig factors ito the equatio of state for a erfect gas. he ader Waals equatio of state is gie by, a P b --- () a or P P = Pressure = olume/uit mass = gas costat b where a ad b are costats for ay oe gas, which ca be determied exerimetally, the costats accout for the itermolecular attractios ad fiite size of the molecules which were a assumed to be o-existet i a ideal gas. he term accouts for the itermolecular forces i.e, force of cohesio ad the term b was itroduced to accout for the olume occuied by the molecules i.e., co-olume. If the olume of oe mole is cosidered, the the aboe equatio ca be writte as, a P b Uits P (N/m ), m / kg mole a 84Nm / kg ( Nm 4 / kg mol b m / kg mol mol 0 K Or 0.084bar m / kg Determiatio of a der Waals costats i terms of critical roerties he determiatio of two costats a ad b i the a der Waals equatio is based o the fact that the critical isotherm o a - diagram has a horizotal iflexio oit at the critical oit. herefore the first ad secod deriatie of P with resect at the critical oit must be zero. mol 0 K Deartmet of Mechaical Egieerig, Page

112 BASIC EMODYNAMICS 0ME i. e., 0ad c c 0 a From equatio () we hae, P b ad b b a 6a 4 At critical oits the aboe equatio reduces to ad b b a 0 6a (a) --- (b) c a Also from equatio () we hae, Pc b Diidig equatio (a) by equatio (b) ad simlifyig we get b c Substitutig for b ad solig for a from equatio (b) we get a = 9 c c --- (c) Substitutig these exressios for a ad b i equatio (c) ad solig for c, we get c b ad 8 c c 8 c c 7 a 64 c c If the olume of oe mole is cosidered the the aboe equatio ca be writte as Deartmet of Mechaical Egieerig, Page

113 BASIC EMODYNAMICS 0ME a P b Uits: P (N/m ), m / kg mol =84 Nm/kg mol 0 K a = Nm 4 / (kg mol) Or bar (m /kg-mole) 0.084bar kg Note: Usually costats a ad b for differet gases are gie. Comressibility Factor ad Comressibility Chart: m mole K b = m /kg-mol he secific olume of a gas becomes ery large whe the ressure is low or temerature is high. hus it is ot ossible to coeietly rereset the behaiour of real gases at low ressure ad high temerature. For a erfect gas, the equatio of state is P =. But, for a real gas, a correctio factor has to be itroduced i the erfect gas to take ito accout the deiatio of the real gas from the erfect gas equatio. his factor is kow as the comressibility factor, Z ad is defied as, Z P Z = for a erfect gas. For real gases the alue of Z is fiite ad it may be less or more tha uity deedig o the temerature ad ressure of the gas. educed Proerties: he real gases follow closely the ideal gas equatio oly at low ressures ad high temeratures. he ressures ad temeratures deed o the critical ressure ad critical temerature of the real gases. For examle 00 0 C is a low temerature for most of the gases, but ot for air or itroge. Air or itroge ca be treated as ideal gas at this temerature ad atmosheric ressure with a error which is <%. his is because itroge is well oer its critical temerature of C ad away from the saturatio regio. At this temerature ad ressure most of the substaces would exist i solid hase. ece, the ressure ad temerature of a substace is high or low relatie to its critical ressure or temerature. Gases behae differetly at a gie ressure ad temerature, but they behae ery much the same at temeratures ad ressures ormalized with resect to their critical temeratures ad ressures. he ratios of ressure, temerature ad secific olume of a real gas to the corresodig critical alues are called the reduced roerties. i. e., P P P c, c & c Deartmet of Mechaical Egieerig, Page

114 BASIC EMODYNAMICS 0ME Law of Corresodig states: his law is used i the aroximate determiatio of the roerties of real gases whe their roerties at the critical state are kow. Accordig to this law, there is a fuctioal relatioshi for all substaces, which may be exressed mathematically as = f (P, ). From this law it is clear that if ay two gases hae equal alues of reduced ressure ad reduced temerature, they will hae the same alue of reduced olume. his law is most accurate i the iciity of the critical oit. Geeralized Comressibility Chart: he comressibility factor of ay gas is a fuctio of oly two roerties, usually temerature ad ressure so that Z = f (, P ) excet ear the critical oit. his is the basis for the geeralized comressibility chart. he geeralized comressibility chart is lotted with Z ersus P for arious alues of. his is costructed by lottig the kow data of oe or more gases ad ca be used for ay gas. It may be see from the chart that the alue of the comressibility factor at the critical state is about 0.5. Note that the alue of Z obtaied from a der waals equatio of state at the critical oit, P c c Z c which is higher tha the actual alue. 8 c Deartmet of Mechaical Egieerig, Page 4