Capter 6 Te Seond Law of Termodynamis In te last two apters of tis book we applied te first law of termodynamis to losed and open systems onsidering bot quasistati and non-quasi-stati proesses. A question tat arises naturally is weter all proesses tat satisfy te first law ould be arried out in pratie. In order to explore tis question we need to introdue te seond law of termodynamis wi teaes us tat tere are additional onditions for te feasibility of a proess. In partiular, te irreversibility of natural proesses is an important onsequene of te seond law. Tere are several statements of te seond law, wi at first appear to be unrelated. However, by treating any one of te different statements of te seond law as a postulate, all te oter statements ould be dedued logially. Te first law predited te existene of a property wi was alled te internal energy of a system. In a similar manner, we sall derive a new property alled te entropy by applying te seond law to a system. In developing te seond law we propose to follow te istorial route tat involves yli proesses and eat engines. From an engineering perspetive, tis approa is deemed to be more pratial and losely related to te subjet matter overed in te preeding apters. 6. Te Heat Engine Cyle One of te main funtions of a majority of energy onversion systems is to generate motive power or work using stored energy from various fuels. Te steam power plant, wi operates on te vapor power yle, is a ommon example of su an energy onversion system. Te various 49
50 Engineering Termodynamis Fig. 6. A vapor-power yle produing a net work output proesses tat onstitute a typial vapor power yle ould be arried out using te piston-ylinder arrangement sown in Fig. 6.. Initially, (Fig. 6.(a)) te well-insulated ylinder ontains a ompressed liquid wit te weigts on te piston generating te desired ig pressure of te yle. Te bottom setion of te insulation around te ylinder is now removed and a eat soure is applied to te ylinder to eat te liquid (Fig. 6.(b)). During tis proess, te liquid first reaes its saturation temperature and ten undergoes pase ange at onstant pressure wile te piston is pused out. After te evaporation proess is omplete, te bottom of te ylinder is insulated and te weigts on te piston are removed in steps to expand te vapor (Fig. 6.()). Te potential energy gained by te weigts onstitutes te external work delivered by te system to te surroundings. Wen te weigt on te piston is redued to an amount tat generates te required low pressure of te yle, te bottom insulation is removed and a old eat sink is applied to te bottom of te ylinder (Fig. 6.(d)). Tis proess is ontinued until all te vapor is ondensed at a onstant pressure. Te eat sink is now witdrawn and te bottom of te ylinder is again insulated. Weigts are ten plaed on te piston in steps to inrease
Te Seond Law of Termodynamis 5 te pressure of te liquid to te required ig pressure, tereby ompleting te yle (Fig. 6.(e)). At te beginning of ea yle of operation, a new set of weigts is plaed on te piston and tese are moved to a iger elevation tereby delivering a net work output to te surrounding. Te eat supplied to onvert te sub-ooled liquid to a vapor onstitutes te energy input to te yle. We used te piston-ylinder arrangement to illustrate te operation of a power yle mainly to relate it to our work in te earlier apters. However, atual vapor power yles are more ompliated in design and tey make use of separate omponents to arry out te various proesses of te yle. Fig. 6. Simple vapor-power yle Te essential sub-omponents of a vapor power yle are depited sematially in Fig. 6.. Ea omponent is a ontrol volume troug wi te working fluid passes periodially. Let us follow a paket of fluid starting from, te entry point to te boiler were te fluid reeives eat from an external eat soure. Te eat soure ould be te furnae atmospere wi is maintained at a ig temperature by burning a fuel like oil or natural gas. In te ase of a nulear power plant, te eat soure is te reator were te fission reation generates te required termal energy. Te
5 Engineering Termodynamis fluid enters te boiler at as a sub-ooled liquid wit a temperature below te saturation temperature orresponding to te pressure of te boiler. As te paket of fluid passes troug te boiler its temperature first inreases to te saturation value and te fluid ten undergoes pase ange to emerge at te exit of te boiler as a saturated vapor. After passing troug te duting te fluid paket enters te turbine at point 3. During its passage troug te turbine te fluid paket expands and leaves at 4, releasing part of its entalpy as external saft work of te turbine. We desribed te operation of a typial turbine in Se. 5.6.. At 5, te fluid enters te ondenser were it rejets eat at onstant pressure to a old eat sink. A typial eat sink is te atmospere or a body of water su as a river. Te fluid paket leaves te ondenser at 6 as a saturated liquid and enters a feed-pump at 7 to be pumped bak to te boiler at. Te feed pump is operated by using a fration of te work produed by te turbine. We see tat ea fluid paket undergoes a yli proess during its passage troug te plant. Te above desription of te vapor power yle is very brief in tat it inludes only te essential details needed to introdue te seond law in te next setion. We notie tat te various proesses undergone by ea paket of fluid in te plant diagram sown in Fig. 6. and te fixed mass of fluid in te piston-ylinder set-up in Fig. 6. are essentially te same. By fousing our attention mainly on te eat and work interations among te system, te eat soure, te eat sink and te surroundings we an draw te energy flow diagram of te eat engine as sown in Fig. 6.3. During ea yle of operation, a quantity of eat Q flows from te eat soure at temperature T to te yli devie wi delivers to te surroundings a net quantity of work W o. A quantity of eat Q is rejeted to te eat sink at temperature T during te yle. Tis simplified diagram is a very useful abstration of te more ompliated systems depited in Figs. 6. and 6.. Terefore, in te disussions to follow in tis apter we sall use te energy flow diagram in Fig. 6.3 to represent a typial eat engine yle.
Te Seond Law of Termodynamis 53 Fig. 6.3 Energy-flow diagram for eat engine 6.. Effiieny of a eat engine yle Te effiieny η of te eat engine yle is defined as te net work output per unit eat input. Tis an be expressed as W Q Applying te first law to te yle we ave From Eqs. (6.) and (6.) we obtain o η (6.) Q Q W (6.) Q Q o η (6.3) It is seen tat te effiieny of te yle as an eonomi signifiane beause in a typial power plant te eat soure is maintained at te ig temperature by burning fuel, wi onstitutes te main operating energy ost of te plant. Te net work delivered to te surroundings, on te oter and, is te desired output of te plant.
54 Engineering Termodynamis Terefore te maximization of te effiieny of te yle sould be te primary objetive of power plant design. 6. Te Reversed Heat Engine Cyle In dealing wit topis related to te seond law we also enounter an important lass of energy onversion systems alled reversed eat engine yles. In pratial terms tese are refrigerators or eat pumps as tey are sometimes alled. Wereas eat engines deliver work by absorbing eat from ig temperature soures, reversed eat engines transfer eat from bodies at low temperatures to bodies at ig temperatures. Te latter proess does not our witout te aid of an external energy input. Fig. 6.4 A reversed eat engine yle Illustrated in Fig. 6.4 is a typial reversed eat engine yle wi operates using a working fluid tat undergoes pase ange. It sould be noted tat tis is an idealized yle tat differs somewat from te ideal vapor-ompression refrigeration yle due mainly to pratial reasons. Let us now follow te passage of a paket of fluid troug te various sub-omponents of te plant sown in Fig. 6.4. Te fluid paket enters te evaporator as a liquid or a wet-vapor at. As it passes troug te evaporator te fluid absorbs eat from te old body beause its
Te Seond Law of Termodynamis 55 temperature is below tat of te old body. In a typial refrigerator te old body is te refrigerated spae. At te fluid is usually a saturated vapor and at 3 it enters te ompressor were its pressure is raised to te ondenser pressure. Tis ompression proess requires an external work input. During te passage troug te ondenser from 5 to 6 te fluid rejets eat to a eat sink, to attain a liquid state at te exit 6. In order for tis eat transfer to take plae te working-fluid temperature in te ondenser as to be iger tan te temperature of te eat sink, wi in te ase of a pratial refrigerator is te atmospere. At 7 te fluid enters an expander wi redues its pressure to te evaporator pressure to omplete te yle. T Heat sink Q Cyli devie Wi Q T Cold body Fig. 6.5 Energy-flow diagram for a reversed eat engine yle Te work produed during te expansion proess ould be made use of to redue te external work input to te ompressor. However, in pratial vapor ompression yles, te expander is replaed wit an expansion valve for pratial reasons tat we will disuss in a later apter. Te proesses of te reversed eat engine yle ould also be arried out using a piston-ylinder set-up similar to tat sown in Fig 6.. As for te eat engine, we an draw te energy flow diagram for te reversed eat engine to indiate te main energy interations as sown in Fig. 6.5.
56 Engineering Termodynamis We note tat te diretions of te eat and work interations for te reversed eat engine are opposite to tose in te eat engine yle. 6.. Coeffiient of performane of a reversed eat engine yle Te performane indiator for te reversed eat engine is alled te oeffiient of performane (COP). In defining te COP we need to identify te desired energy interation, wi is te main funtion of te reversed eat engine and te input energy form. Two situations need to be distinguised for tis purpose. Wen te objetive of te reversed engine is to extrat eat from te old body using te work input we all te devie a refrigerator, te COP of wi is defined as Applying te first law to te yle we ave From Eqs. (6.4) and (6.5) we obtain Q ( COP ) r (6.4) Wi Q Q W (6.5) Q i ( COP) r (6.6) Q Q It is interesting to note tat we an use te reversed eat engine to transfer eat to a otter body by extrating eat from a soure su as te atmospere or te ground wi is at a lower temperature. Su a devie is alled a eat pump beause te desired purpose of te devie is to transfer eat to te otter body. In reent years eat pumps ave found wide spread appliation in te eating of omes and buildings. Te COP of te eat pump is defined as Applying te first law to te yle we ave Q ( COP ) (6.7) Wi Q Q W (6.8) i
Te Seond Law of Termodynamis 57 From Eqs. (6.7) and (6.8) we obtain From Eqs. (6.6) and (6.9) it follows tat Q ( COP) (6.9) Q Q ( COP ) + ( COP) (6.0) Having disussed te main features of eat engines and reversed eat engines we now pose te following questions onerning te design and operation of su devies. (i) Is te ondenser in te vapor power yle sown in Fig. 6. really neessary? After all it absorbs a portion of te energy from te eat soure wi ould oterwise ave been used to produe more work. (ii) Is tere an upper limit to te effiieny of te eat engine yle? (iii) Are tere proesses and onditions tat ave an adverse effet on te performane of eat engine yles? (iv) In a reversed eat engine, is it really neessary to ave a work input to transfer eat from a old body to a ot body? (v) Is tere an upper limit to te COP of a reversed eat engine? From a istorial perspetive, it was te sear for te answers to tese interesting and relevant questions tat lead to te formulation of te seond law of termodynamis. r 6.3 Te Seond Law of Termodynamis Te seond law of termodynamis as been stated in many different forms, some of wi appear at first to ave no relation to te oter forms. If we aept any one of te statements of te law as a postulate, ten te oter forms an be proved by using logial arguments. However, te statement of te seond law tat we start wit annot be derived from any oter law of nature. Two of te well-known statements of te seond law, alled te Kelvin-Plank statement and te Clausius statement, are diretly related to eat engines and reversed eat engines respetively. Te Kelvin-Plank statement (K-P-S): It is impossible to onstrut a devie tat will operate in a yli manner and produe no effet oter
58 Engineering Termodynamis tan produe work wile exanging eat wit bodies at a single fixed temperature. We an elaborate te pratial relevane of tis statement by referring to te eat engine yle sown in Fig. 6.. In tis ase, te omponents witin te boundary indiated by te broken-lines, is te yli devie. Heat flows to te devie from te ig temperature soure and tere is a net prodution of work. Te Kelvin-Plank statement states tat it is impossible for tis eat engine yle to operate if a fration of te eat input is not rejeted to te eat sink. In te absene of te eat sink, te eat flow from te eat soure is equal to te work prodution by virtue of te first law. Terefore te effiieny of te eat engine is 00 perent aording to Eq. (6.). Tus far, all efforts to onstrut a yli devie tat onverts all te eat reeived from a soure at a single temperature ontinuously to work ave failed. Su an engine is alled a perpetual motion maine of te seond kind. Te Clausius statement (C-S): It is impossible to onstrut a devie tat operates in a yli manner and produes no effet oter tan te transfer of eat from a older body to a otter body. Tis statement as a diret bearing on te operation of reversed eat engine yles of te type sown in Fig. 6.4. Consider te omponents witin te broken-line boundary in Fig. 6.4 as te yli devie. Te Clausius statement states tat it is impossible for tis reversed eat engine yle to operate if te external work input to te devie is witdrawn. In te absene of te work input, eat entering from te old body is equal to te eat transferred to te ot body by virtue of te first law. Moreover, te COP of te yle is infinity aording to Eq. (6.4). Terefore aording to te Clausius statement, it impossible to onstrut a refrigerator or eat pump tat operates in a yle manner witout an input of work. Also, te COP of te refrigerator always as a finite value.
Te Seond Law of Termodynamis 59 6.3. Equivalene of te Kelvin-Plank and Clausius statements We sall now prove tat te K-P-S and C-S of te seond law are equivalent. Tat is if a devie wi violates te C-S an be onstruted ten te devie also violates te K-P-S and vie versa. Sown in Fig. 6.6 is a yli devie CS wi transfers steadily an amount of eat Q from a old body to a ot body wit no work input. Su a devie violates te C-S. We now operate a eat engine HE between te ot body and te old body, wose design is su tat it reeives an amount ( Q + W ) of eat from te ot body, produes an amount W of work and rejets an amount Q of eat to te old body. Consider te units witin te boundary indiated by te broken-lines as a omposite yli devie. It reeives a net amount of eat W from te ot body and overts all of it to work tus violating te K-P-S. To prove te onverse onsider te yli devie KPS sown in Fig. 6.7. It reeives an amount Q of eat from te ot body and onverts all of it to work, violating te K-P-S. We now operate a eat pump HP between te two bodies using te work input from KPS. Te eat pump extrats an amount Q of eat from te old body and transfer an amount ( Q + W ) of eat to te ot body. Consider te omposite yli devie witin te broken-line boundary. T Q W Q +W KPS W HP Q T Fig. 6.6 Equivalene of CS and KPS Fig. 6.7 Equivalene of KPS and CS
60 Engineering Termodynamis Tis devie extrats an amount Q of eat from te old body and transfers te same amount of eat on a net basis to te ot body wit no external work input, tereby violating te C-S. Any devie tat violates te C-S will violate te K-P-S and terefore te seond law of termodynamis. Su a devie is alled a perpetual motion maine of te seond kind or PMM. Any devie tat produes an energy output larger tan te energy input will violate te first law of termodynamis. Su a devie is alled a perpetual motion maine of te first kind or a PMM. 6.4 Reversible and Irreversible Proesses Reversible proesses play an important role in deduing some important onsequenes of te seond law. A reversible proess is a proess wi after it as ourred an be reversed witout leaving any ange eiter in te system or te surroundings. We now examine te reversibility of some natural proesses in ligt of te above definition. Te C-S of te seond law is a formalization of our everyday experiene tat eat flows unaided from a body at a iger temperature to a body at a lower temperature (Fig. 6.8). Te reversal of tis natural proess, owever, requires te use of a eat pump wit an external work input wi as to be positive aording to te C-S. Terefore te reversal of te proess will leave a permanent ange in te surroundings due to te work extrated. Tere are numerous oter proesses tat seem to always follow a preferred diretion naturally, altoug te first law would allow tese proesses to our bot in te forward as well as te reverse diretions. Let us onsider tree su proesses to wi we applied te first law earlier in apter 4. Te well-insulated vessel sown in Fig. 6.9 is divided into two ompartments by a diapragm wit one ompartment evauated and te oter ontaining a gas. If te diapragm ruptures te gas will expand to fill te wole vessel.
Te Seond Law of Termodynamis 6 T ot T old Gas Vauum Fig. 6.8 Heat transfer irreversibility Fig. 6.9 Free expansion To reverse tis proess, we need to ompress te gas using a work input from te surroundings wile transferring eat to a old body to ontrol te temperature of te gas. Tis way te gas ould be returned to its initial state but tere is a permanent ange in te surroundings due to te work extrated and te eat transferred to te old body. To restore te surrounding to its original state, te eat transferred to te old body will ave to be ompletely reonverted to work. Te latter proess, owever, would violate te K-P-S. Te paddle weel sown in Fig. 6.0 is immersed in water ontained in an insulated tank. Wen te weigt attaed to te rope is released te rotation of te paddle weel tat follows will inrease te temperature of te water due to fritional effets. To reverse tis proess we ould, in priniple, operate a yli eat engine wit te water as te eat soure and a old body as te eat sink. Te work output of te engine ould be used to partially lift te weigt. Sine some eat as to be rejeted to te eat sink, as required by te K-P-S, all te work originally transferred by te paddle weel to te water annot be extrated. Terefore, an additional work input will be required to lift te weigt to te original eigt. Altoug te water and te weigt ave been returned to teir initial states, permanent anges ave ourred in te surroundings. Tese anges, owever, annot be reversed witout violating te K-P-S. In te tird example we onsider te two different gases ontained in te two ompartments of te vessel sown in Fig. 6.. If te diapragm separating te two setions ruptures, te gases will expand to fill wole vessel. Te reversal of tis proess will require a omplex gas separation proess tat will leave permanent anges in te surroundings.
6 Engineering Termodynamis WEIGHTS Fig. 6.0 Fritional eating Fig. 6. Irreversible mixing of gases We an summarize te ommon arateristis of te various proesses desribed in te above examples as follows: (i) Te initial equilibrium state of te system is maintained by te onstraints on te system. In te first and tird examples (Figs. 6.9 and 6.) tey are te separating diapragms wile te platform olding weigt is te onstraint in te seond example (Fig. 6.0). (ii) Wen te onstraint is removed suddenly, te system passes troug a series of non-equilibrium states involving rapid dynami anges before te final equilibrium state is attained. (iii) Te system an only be returned to te initial state by energy exanges wit te surroundings tat leave permanent anges in te latter if te seond law is not to be violated. Te question tat follows immediately is ow do we arry out a proess reversibly? We ave disussed te answer to tis question partially wen we onsidered quasi-equilibrium proesses in Capter. At any stage in a quasi-equilibrium proess te onstraints on te system are ontrolled in su a manner tat deviation from equilibrium is infinitesimal. Tis requires tat tere sould be negligible intensive - property gradients witin te system. Te unbalane of te fores at te boundary of te system sould be negligible and eat transfer between te system and te surroundings sould our only troug an infinitesimal temperature differene. We disussed two su proesses in some detail in te worked examples.5 and.6 in apter.
Te Seond Law of Termodynamis 63 Terefore a neessary ondition for a proess to be reversible is tat it sould be arried out in a quasi-stati manner. Te only additional ondition for reversibility is te omplete absene of solid or fluid frition effets during te proess. Fritional effets wen present onvert some of te work input to a eat interation. Aording to te seond law tis eat annot be ompletely onverted bak to work using a eat engine beause su a devie would violate te K-P-S. Terefore te presene of fritional effets renders a proess irreversible. In summary a proess will be reversible wen (i) it is performed in a quasi-stati manner and (ii) it does not involve any fritional effets. Worked examples 6., 6. and 6.3 disuss te reversibility of several proesses of pratial importane. 6.4. Types of irreversible proesses Consider te vapor power yle sown in Fig. 6.. Assume tat te proesses undergone by te working fluid in te boiler our in a quasistati manner witout fritional effets. However, let tere be a finite temperature differene between eat reservoir and te fluid in te boiler wen eat is transferred to te latter. We all tis situation an external termal irreversibility beause te irreversible proess ours outside te system, wile te system itself undergoes a reversible ange. Similarly, te onversion of te work input of te paddle weel to internal energy of te water due to fritional effets (Fig. 6.0) is an external meanial irreversibility. In tis ase, te fritional irreversibility ours at te boundary between te paddle weel and te water wi is te system of interest. Here we assume tat te temperature and pressure gradients in te water are negligible. Te gas expansion following te rupture of te diapragm in Fig. 6.9 is an internal meanial irreversibility. Te irreversible proess, wi in tis ase, is te free expansion of te gas aross a finite pressure differene, ours inside te system. An internal termal irreversibility is a eat flow proess tat takes plae due to a finite temperature differene witin a system as in Fig. 6.8. Te mixing of te two gases sown in Fig. 6. is alled a emial irreversibility.
64 Engineering Termodynamis 6.4. Reversible eat engines and termal reservoirs Te reversible proess is a useful idealization of real proesses tat enables us to analyze tem onveniently beause te properties of te system are spatially uniform at every stage. We now extend tis onept to eat engine yles by defining a reversible eat engine yle as a yle in wi all te proesses are reversible. Terefore te yle an be ompletely reversed leaving no permanent effets on te surroundings. Te eat and work interations of te reversed yle are equal in magnitude but opposite in diretion to te orresponding interations in te diret yle. A termal or eat reservoir is an idealized eat soure or sink tat enables us to arry out eat transfer proesses in a reversible manner. It is defined as a body wose mass is su tat te absorption or rejetion of a quantity of eat of any magnitude will not result in an appreiable ange in its temperature or any oter termal property. We define an ideal eat engine yle as a reversible eat engine operating between a ig temperature reservoir and a low temperature reservoir. Tis yle is reversible bot internally and externally. 6.5 Some Consequenes of te Seond Law We dedued from te K-P-S of te seond law tat te termal effiieny of any eat engine as to be less tan 00 perent. A question tat follows diretly is wat is te maximum possible effiieny tat a eat engine an ave? Te answer to tis question was provided by a Fren engineer, Niolas Leonard Sadi Carnot in 84. He developed a reversible yle, ommonly known as te Carnot yle, tat ould be arried out wit a piston-ylinder arrangement using an ideal gas as te working fluid. Te expression for te effiieny of te Carnot yle, owever, is independent of te detailed design of te eat engine and terefore it sould be appliable to all reversible yles. We sall now prove te latter statement, ommonly known as te Carnot priniple, using te K-P-S of te seond law. Te Carnot priniple states tat:
Te Seond Law of Termodynamis 65 (i) No eat engine an be more effiient tan a reversible engine operating between te same ig temperature eat reservoir and te low temperature eat reservoir. (ii) Te effiieny of all reversible eat engines operating between te same ig and low temperature reservoirs is te same. Fig. 6. Te Carnot priniple for eat engines We sall prove te first statement by omparing te effiieny of any engine EX and a reversible engine ER by making use of te arrangement sown in Fig. 6.. Here te reversible engine is reversed and operated as a reversible eat pump. Te engine EX reeives a quantity of eat Q from te ig temperature reservoir and produes a quantity of work Wx as its output. Te orresponding quantities of eat and work for te engine ER are Q and W r respetively. Let us assume tat te effiieny of EX is larger tan te effiieny of ER. Terefore W x > Wr beause te eat input for bot engines is te same. Now wen ER is reversed and made to funtion as a eat pump, te magnitude of te eat and work interations remain te same but tey our in te opposite diretions.
66 Engineering Termodynamis Terefore te reversed-er will transfer Q to te ig temperature reservoir using a work input of W r. Tis work input is supplied by EX leaving a quantity of work ( Wx Wr ) for delivery to te surroundings. Consider te omposite yli devie enlosed witin te broken-line boundary. It reeives a quantity of eat ( Wx Wr ) from te low temperature reservoir and onverts all of it to work on a net basis, violating te K-P-S of te seond law. Terefore our assumption tat te effiieny of EX is larger tan te effiieny of ER is invalid. We onlude tat no engine an ave an effiieny larger tan tat of a reversible engine operating between te same reservoirs. Te seond statement an be proved by following a similar type of reasoning as for statement. Consider te arrangement in Fig. 6. were EX is now a seond reversible engine wose effiieny is iger tan tat of ER. We reverse te engine ER wit te lower effiieny and operate it as a reversible eat pump wit te work input supplied by engine EX wit te iger effiieny. Te omposite devie witin te broken-line boundary will absorb eat from te low temperature reservoir and onvert all of it to work tus violating te K-P-S. Te same reasoning will apply no matter wi of te two engines is assumed to ave te iger effiieny. Te onlusion terefore is tat bot reversible engines must ave te same effiieny tus proving part (ii) of te Carnot priniple. By following a similar proedure (see problem 6.4) we ould prove (i) tat no eat pump (reversed eat engine) ould ave a COP iger tan tat of a reversible eat pump and (ii) tat all reversible eat pumps must ave te same COP. 6.5. Effiieny of a Carnot yle using an ideal gas We now derive an expression for te effiieny of a partiular reversible eat engine operating wit a piston-ylinder arrangement were te working fluid is an ideal gas. Te generalization of tis expression for any reversible engine will be disussed in te next setion.
Te Seond Law of Termodynamis 67 Consider a fixed mass of an ideal gas ontained in a piston-ylinder arrangement similar to tat sown in Fig. 6.. Te gas is brougt into ommuniation wit a ig temperature reservoir at temperature T wose temperature differs infinitesimally from tat of te gas. Work is done by te gas during tis proess and onsequently its temperature is maintained onstant. Tis proess is sown as - in te P-V and T-V diagrams of Figs. 6.3(a) and (b) respetively. During te proess -3 te ylinder is insulated and te gas is subjeted to a reversible adiabati expansion delivering work to te surroundings. A low temperature eat sink at T is ten applied to te ylinder wile te gas is ompressed isotermally as indiated by proess 3-4 in te figures. Tere is only an infinitesimal differene between te temperatures of te eat sink and te gas to ensure tat te proess is reversible. Finally, te gas is subjeted to a reversible adiabati ompression from 4 to to restore its initial state, tus ompleting te yle. Applying te general expression in Eq. (6.3) we obtain te effiieny of te yle as Q Q η (6.) were Q is te eat reeived from te ig temperature reservoir during proess - and Q is te eat transferred to te low temperature reservoir during proess 3-4. Tere is no eat transfer during te two adiabati proesses -3 and 4-. Hene Q 3 0 (6.) Q 4 0 (6.3) Te ange in internal energy of an ideal gas during an isotermal proess is zero. Hene by applying te first law to te isotermal proesses - and 3-4 we ave Q W and W34 Q (6.4)
68 Engineering Termodynamis Fig. 6.3(a) Carnot yle: P-V diagram Fig. 6.3(b) Carnot yle: T-V diagram Te expression for te work done by an ideal gas during an isotermal proess was obtained earlier in worked example 3. in apter 3. Applying te expression to proesses - and 3-4 we obtain te following W W ( V V ) mrt ln / (6.5) ( V V ) 34 mrt ln 3 / From Eqs. (6.) (6.6) it follows tat (6.6) 4 Q T ln( V3 / V4 ) η (6.7) Q T ln( V / V ) In worked example 4.9 in apter 4 we sowed tat te P-V relation for a quasi-stati adiabati proess of an ideal gas to be γ PV C (6.8) were C is a onstant and γ is te ratio of te speifi eat apaities. Te ideal gas equation of state is PV mrt (6.9) Eliminating P between Eqs. (6.8) and (6.9) we obtain te following relation for a quasi-stati adiabati proess of an ideal gas were C is a onstant. TV γ C (6.0)
Te Seond Law of Termodynamis 69 Applying Eq. (6.0) to te adiabati proess -3 and 4- we ave γ γ 3 T V TV (6.) γ γ 4 T V TV (6.) Dividing Eq. (6.) by Eq. (6.) V / V V V (6.3) 3 / Substituting from Eq. (6.3) in Eq. (6.7) we obtain te expression for te effiieny as T η (6.4) T Altoug te derivation of te Eq. (6.4) was somewat tedious, te final expression, owever, involves only te absolute temperatures of te two reservoirs. It is noteworty tat te expression for te effiieny of te Carnot yle is independent of te design parameters of te pistonylinder set-up and te operating onditions of te yle. In te next setion we sall obtain an expression similar to Eq. (6.4) following a different route tat leads us to te onept of a termodynami temperature sale. 4 6.5. Termodynami temperature sale An important onsequene of te reversible eat engine is te development of a termodynami temperature sale (Absolute temperature sale) wi is independent of te termometri properties of materials. Te Carnot priniple sowed tat te effiieny of all reversible eat engines operating between te same two reservoirs is te same. Terefore te effiieny of te eat engine sould only be a funtion of te two reservoir temperatures. In te preeding setion, we demonstrated tis for a partiular reversible eat engine. Hene te effiieny of a reversible engine may be expressed as η Q / Q F( T, T ) (6.5) It follows tat Q Q F( T, T ) f ( T, T ) (6.6) / were F and f are funtions of te two reservoir temperatures.
70 Engineering Termodynamis Fig. 6.4 Termodynami temperature sale Consider te arrangement of reversible eat engines sown in Fig. 6.4 were te engine ER operates between reservoirs of temperatures T and T wile te engine ER operates between reservoirs of temperatures T and T 3. A tird reversible engine ER3 operates between reservoirs at temperature T and T 3. Applying Eq. (6.6) to ea eat engine in turn we obtain te following relations. Consider te relation Q Q f ( T, ) (6.7) / T Q Q f ( T, ) (6.8) 3 / T3 Q Q f ( T, ) (6.9) 3 / T3 ( Q / Q )( Q ) Q / Q (6.30) 3 Q 3 / Substituting from Eqs. (6.7) (6.9) in Eq. (6.30) we ave f T, T ) f ( T, T ) f ( T, ) (6.3) ( 3 T3 Now te left and side of Eq. (6.3) is independent of T. Terefore f T, T ) and f T, T ) sould ave te following funtional form ( ( 3 f ( T, T ) φ( T ) / φ( T )
Te Seond Law of Termodynamis 7 and f T, T ) φ( T ) / φ( ) ( 3 3 T In general, f ( T, T ) sould be of te form f T, T ) φ( T ) / φ( T ) (6.3) ( were T and T are te temperatures of te ot and old reservoirs respetively. Substituting in Eq. (6.6) we ave Q φ( T ) / φ( T ) (6.33) / Q Tere are several forms of te funtion, φ (T ) wi will satisfy Eq. (6.33). In 854 Tomson (Lord Kelvin) proposed te simple form, φ ( T ) T, wi was osen to agree wit te ideal gas temperature sale. Tis new sale, wi is now universally used, is alled te termodynami temperature sale (absolute temperature sale) or te Kelvin temperature sale. Substituting in Eq. (6.33) for te funtion φ (T ) we ave Q / Q T / T (6.34) Substituting in Eq. (6.5) from Eq. (6.34) we obtain te effiieny of te Carnot engine as η T / T (6.35) In order to define te absolute temperature ompletely we need to fix te size of a degree in te termodynami temperature sale. Te Tent International General Conferene on Weigts and Measures in 954 agreed on te following definition: Te Kelvin unit of termodynami temperature is te fration /73.6 of te termodynami temperature T tp of te triple point of water. Terefore te numerial value assigned to te temperature of te triple point of water is 73.6 K. Consider te tougt-experiment were a Carnot eat engine is operated between a reservoir at T, te temperature to be measured, and a reservoir at T tp, te triple-point temperature of water. Let te respetive eat transfers per yle between te reservoirs and te eat engine be Q and Q tp. If te measured effiieny of te Carnot engine is η tp, ten it follows from Eqs. (6.34) and (6.35) tat ( ) T T Q / Q T / η tp tp tp tp
7 Engineering Termodynamis We observe tat te above equation defines a temperature sale tat is independent of termometri substanes and material properties of temperature sensors. It depends, in onept, only on te effiieny of te reversible eat engine. Te Celsius temperature is ten defines by T( o C) T(K) 73.5 From te analysis of a reversible Carnot yle, using te ideal gas temperature defined in Se..5.3, we obtained Eq. (6.4) for its termal effiieny. Tis expression is similar to te expression in Eq. (6.35) tat is based on te termodynami temperature sale. It is te oie of te funtion, φ ( T ) T and te numerial mating of te temperatures at te triple point of water tat as made te ideal gas temperature sale and te termodynami temperature sale idential. Note, owever, tat wereas to use te Celsius degree requires measurements at two fixed points (typially te ie-point and te steam-point) to use te degree defined in te new way requires measurements only at one fixed point (te triple-point of water). 6.5.3 Cyles interating wit a single termal reservoir We now summarize in analytial form some of te important onsequenes of te seond law for yli devies operating wit a single termal reservoir. Su a devie is sown in Fig. 6.5. From te K-P-S it follows tat for te yle i yle W 0 (6.36) were W i te work output of proess i of te yle wi is taken as positive. Applying te first law to te yle we ave yle Q W 0 (6.37) i yle Te integral forms of Eqs. (6.36) and (6.37) are i δw 0 and δq 0 (6.38)
Te Seond Law of Termodynamis 73 Now te K-P-S of te seond law does not distinguis expliitly between reversible and irreversible yles. It is lear tat for an irreversible yle, te net eat input and te net work output are bot negative aording to te K-P-S. Now onsider a reversible eat engine operating as a eat pump, interating wit a single eat reservoir. Te eat and work interations are sown by te broken-line arrows in Fig. 6.5. Su a devie is perfetly feasible beause work an be ompletely onverted to eat using a fritional devie like a paddle weel. If we now reverse te eat pump sown in Fig. 6.5, te reversibility of te yle will ensure tat te eat and work interations are reversed wit exatly te same magnitudes and no oter effet on te surroundings. Tis reversible eat engine yle, owever, violates te K-P-S beause it onverts te entire eat input to work. Terefore we onlude tat for te K-P-S and reversibility to be satisfied simultaneously, te eat and work interations must bot be zero for a reversible yli devie exanging eat wit a single reservoir. Heat reservoir Q ER W Fig. 6.5 Cyli devie operating wit a single eat reservoir A useful test to determine weter a given proess, in wi eat is exanged wit a single reservoir, is irreversible or oterwise emerges from Eqs. (6.36) and (6.37) and teir yli integral forms in Eq. (6.38). Te proedure an be stated as follows. Consider te end-states of te proess being tested for irreversibility and introdue a reversible proess tat would take te system from te final state bak to te initial state exanging eat wit te same
74 Engineering Termodynamis reservoir. If te net work transfer of te yle onsisting of te original proess and te new reversible proess is negative ten te original proess is irreversible aording to Eq. (6.37). If, owever, te net work transfer is zero ten te original proess is reversible. It is lear tat if te net work transfer is positive, te K-P-S is violated. Te appliation of te above proedure is demonstrated in worked examples 6.4 to 6.9. 6.5.4 Cyles interating wit two termal reservoirs Consider te reversible eat engine operating between two termal reservoirs sown sematially in Figs. 6.3 and 6.5. Applying te first law using te sign onvention adopted in apter 4 we ave te following onditions. For a eat engine: i yle For a reversed eat engine: i yle W > 0 and Q > 0 i W < 0 and Q < 0 were and denote te two reservoirs. As a onsequene of te seond law we obtained an additional relation given by Eq. (6.34) for a reversible yle wit two eat reservoirs. Using te sign onvention in apter 4 we an rewrite Eq. (6.34) in te form i Q T i i 0 i i i (6.39) for bot eat engines and reversed eat engines. Te Carnot priniple states tat no engine operating between two eat reservoirs an ave an effiieny greater ten a reversible engine. Terefore η < η (6.40) irr rev
Te Seond Law of Termodynamis 75 Substituting te expressions for te effiieny of an irreversible yle from Eq. (6.3) and for a reversible engine from Eq. (6.35) in te above inequality we obtain te ondition Q i < (6.4) Q i were te sub-sript i indiates te eat interations for te irreversible yle. Qi Qi Hene < 0 T T T T (6.4) Note tat we ould sow tat te ondition given by Eq. (6.4) also applies to reversed eat engines or eat pumps by using te fat tat COP irr < COP rev Combining Eq. (6.39) and te inequality (6.4) we ould onlude tat for reversible and irreversible eat engines and eat pumps i Q T i i 0 (6.43) In applying Eq. (6.43), appropriate signs must be used for te two eat interations as tese are different for eat engines and eat pumps. In summary, to be feasible, a yli devie operating between two eat reservoirs must satisfy Eq. (6.43). Oterwise te devie violates te seond law. Having obtained te onditions resulting from te seond law for eat engines operating wit one and two reservoirs, te next logial step is to apply te seond law to reversible and irreversible yli devies operating wit any number of termal reservoirs. 6.5.5 Cyles interating wit any number of termal reservoirs To apply te seond law to a system tat exeutes a yle exanging eat wit any number of reservoirs we onsider te arrangement sown in Fig. 6.6.
76 Engineering Termodynamis RESERVOIR To Qoi Wi Qi Ti RESERVOIRS Qsi T W SYSTEM, S Fig. 6.6 Cyli devie operating wit any number of reservoir A losed system S interats wit a series of reservoirs at T i, wi in turn reeive eat from a single reservoir at T o. Te yle exeuted by te system and te eat transfer interations between te system and te various eat reservoirs ould be reversible or irreversible. However, te eat transfer between te reservoir at T o and ea of te seondary eat reservoirs at T i are reversible beause eat transfer is arried out by operating a reversible eat engine as sown in Fig. 6.6. Te eat engines deliver work to te surroundings wile te required eat transfer ours. Te temperature T, at te loation on te system were eat is exanged wit te seondary reservoir at T i, may or may not be equal to te latter temperature. If T and T i are different ten te system would experiene an external eat transfer irreversibility in addition to any irreversibilities tat are internal to te system S. Assume tat te eat engines exeute an integral number of yles per yle of te system wose work output is W per yle. Te eat input to and te work output from te various engines are Q oi and W i per engine yle.
Te Seond Law of Termodynamis 77 Now for ea of te reversible eat engines we apply Eq. (6.39) to obtain Q T oi o Q T i i 0 Sine ea of te seondary termal reservoirs operates in a yle (6.44) Q i Q si (6.45) Consider te yli devie enlosed witin te broken-line boundary in Fig. 6.6. Te total work output of te devie per yle of te system is W W W (6.46) d i + yle Te first term on te rigt and side of Eq. (6.46) is te total work done by te reversible engines per yle of te system and te seond term is te work output of te system. Te devie reeives eat from a single reservoir at T o and onverts all of it to work violating te K-P-S of te seond law. Terefore applying Eq. (6.36) we ave W Wd Wi + 0 (6.47) yle Apply te first law to te yli devie and invoke Eq. (6.47) to obtain Substituting for yle Q oi yle W i + W 0 (6.48) Q oi from Eq. (6.44) in Eq. (6.48) we ave Qi T o 0 (6.49) T Sine T o is positive, te relation between te eat reeived by te reservoir and its temperature beomes Qi 0 T i i (6.50)
78 Engineering Termodynamis Substituting from Eq. (6.45) in Eq. (6.50) we ave Qsi 0 T i (6.5) In te above equation te inequality sign applies wen te proesses undergone by te system S are irreversible. Te irreversibilities ould inlude bot internal irreversibilities of te system and external irreversibilities due to te eat transfer aross finite temperature differenes between te reservoirs at T i and te loation on te system at T. Wen all te proesses are reversible te equality sign applies. If te yli proess does not satisfy te ondition expressed by Eq. (6.5) ten it violates te seond law. 6.6 Worked Examples Example 6. Propose ideal reversible proesses to reverse te proesses listed below. Hene sow tat te original proesses are irreversible. (i) A blok of metal at 0 o C is plaed in termal ontat wit a eat reservoir at 30 o C until teir temperatures are equal. (ii) A well-insulated ylindrial vessel is divided into two ompartments wit a ligt insulating piston. Initially a fixed mass of air at 80 o C and bar pressure is trapped in one setion wile te oter setion is evauated. Te piston is eld in position by a pin. Te pin is removed and te air expands to fill te wole vessel. (iii) A well-insulated piston-ylinder apparatus ontains wet steam at a temperature of 05 o C and quality of 0.3. Te external pressure on te piston due to te weigt of te piston and te surrounding air pressure is onstant. Loated inside te ylinder is an eletrial-resistane eater. Te eater is swited on and te steam evaporates until te quality is 0.4. Solution (i) In te final equilibrium state te metal blok will ave te same temperature of 30 o C as te reservoir. We operate a reversible or Carnot eat pump to transfer eat from te reservoir to te blok to raise its temperature to 0 o C tus ompleting te yle exeuted by te omposite system onsisting of te blok and eat pump.
Te Seond Law of Termodynamis 79 Te net work interation of tis yle wi exanges eat wit a single reservoir at 30 o C is negative beause of te work input to te eat pump. Terefore te yle is irreversible aording to Eq. (6.36). Sine te seond proess introdued by us is reversible, te original proess as to be irreversible. (ii) Wen te pin is removed, te ligt piston will be pused out and te air will expand to fill te vessel. For tis proess te work done is zero beause of te free expansion and te eat interation is zero due te insulation. Terefore aording to te first law te internal energy is unanged. Treating te air as an ideal gas we onlude tat te temperature of te air at te final state is also 80 o C. In order to reverse te proess we ompress te air in a quasi-stati isotermal proess to te original volume maintaining termal ontat wit a reservoir at 80 o C. Te net work interation of te yle is negative beause of te work of ompression and terefore te yle irreversible aording to Eq. (6.36). Sine te ompression proess is reversible, te original free-expansion proess as to be irreversible. (iii) Te proess an be reversed by removing some of te insulation and plaing a reservoir at 05 o C in termal ontat wit te steam. An infinitesimal temperature differene between te steam and te reservoir will initiate ondensation. Te ondensation proess at onstant pressure and temperature is ontinued until te quality of te steam is 0.3 tereby returning te steam to its original state. Te work done by te piston during te expansion is equal in magnitude and opposite in diretion to te ompression work during ondensation. However, te eletrial energy input during original evaporation proess is a negative work interation wi makes te net work interation of te yle negative. Terefore te yle is irreversible aording to Eq. (6.36). Te ondensation proess is reversible and terefore te original eating proess is irreversible. Example 6. A mass M attaed to a spring and a damper rests on a table between two guides as sown in Fig. 6.. Te damping fore is proportional to te relative veloity between te piston and te ylinder and te spring fore is proportional to te displaement.
80 Engineering Termodynamis (i) Desribe a metod to move te mass in a reversible manner from positions A to B onsidering te following onditions (a) te damper is detaed from te mass and tere is no frition between te mass and te guides, (b) te damper is attaed and tere is no frition between te mass and te guides and () te above situations wit sliding frition between te mass and te guide. (ii) Wen te mass is at B te onstraints tat ensured equilibrium during te quasi-stati proess are suddenly removed. Desribed te ensuing motion of te mass and loate its final equilibrium position for ases (a), (b) and () above. Fig. E6. Displaement of a sliding mass Solution (a) In order to move te mass from A to B in a quasi-stati manner we atta to it a ord tat runs over a fritionless pulley as sown in Fig. E6.. Weigts are ung at te end of te ord in small steps so tat we ould ontrol te motion to any degree we desire. If a small weigt is removed, te diretion of motion is reversed due to te tension in te spring leaving no permanent ange in te system or te surroundings. Now for equilibrium T mg kx were T is te tension, x is te displaement and k is te spring onstant. Terefore for small anges of mass, gδ m kδx. Wen te mass is at position B and te ord is ut, te tension in te spring will not be balaned any longer. Te mass will exeute osillations about position A were te spring fore is zero. Ideally, in
Te Seond Law of Termodynamis 8 te absene of any dissipative meanisms, su as air resistane, te osillations will ontinue steadily. (b) Te situation is similar to (a) above exept tat te damper is now attaed. Te fore balane on te mass gives, T mg kx + v were v is te veloity of te mass and is te fore per unit veloity of te damper. Sine mass is moved troug infinitesimal steps quasistatially, its veloity is zero. Consequently, te damping fore is also zero. Te position of te mass an terefore be reversed quasi-statially as in (a). However, wen te ord is ut, te mass will gain kineti energy and te damper will exert a fore opposing te motion. In te damper te work done by te damping fore is onverted to a eat interation tereby inreasing its termal internal energy. Tis proess is irreversible. Te mass will exeute damped osillations about position A until it omes to rest at A wen all te stored meanial energy onsisting of te strain energy of te spring and te kineti energy of te mass is onverted to internal energy of te damper. () Te situations wit regard to (a) and (b) are quite different wen sliding frition is present. As indiated in Fig. E6., te fritional fore between te sliding surfaes as a limiting value given by F µ R µ Mg Tis maximum value is reaed wen te mass is just about to move. Te equilibrium equation for te mass wen it is moved towards position B is given by T kx + µ Mg Terefore in order for te mass M to move, te tension T in te ord as to exeed te spring fore, kx by µ Mg. On te oter and, to reverse te diretion of movement of M, te spring fore as to exeed te tension T by µ Mg beause te fritional fore F anges diretion to oppose te motion. Terefore small anges in te tension of te ord will not produe small anges in te displaement of M in bot diretions as in te ase of (a) and (b). Moreover, any movement of M will involve some fritional work at te sliding surfae wi is onverted to a eat interation. Tis proess is learly irreversible.
8 Engineering Termodynamis Terefore wen sliding frition is present, it is not possible to move te mass M from A to B in a reversible manner. At position B, wen te ord is ut te mass M will exeute damped osillations wit meanial energy dissipation in te damper and at te sliding surfae. Eventually te mass will ome to rest at a position tat is witin ( ± µmg / k) about A wen te spring fore balanes te fritional fore. Example 6.3 Figure E6.3 sows an arrangement for te onstantvolume eating of an ideal gas A in a reversible manner by supplying work to te ideal gas B trapped in a piston-ylinder set-up. Te wall separating te onstant volume of gas A from gas B is rigid and made of a igly onduting material. Te piston and te ylinder are wellinsulated on te outside. (a) Obtain an expression for te P-V relation for te gas B in terms of te quantities indiated in te figure and (b) suggest two oter metods for eating a gas reversibly at onstant volume. Solution Te various relevant quantities like te temperatures, pressures and volumes are indiated in Fig. E6.3. Applying te first law to te omposite system onsisting of te two gases we ave δ Q du + δw (E6.3.) Te system is insulated and te ompression proess is quasi-stati. Due to te igly onduting wall between gases A and B teir temperatures are equal. Let tis temperature be T. Wit tese onditions, Eq. (E6.3.) beomes 0 ( m + m ) dt + P dv (E6.3.) were v is te speifi eat apaity of te gas. Applying te ideal gas equation to gas B B vb A va B PB VB mbrbt (E6.3.3) From Eqs. (E6.3.) and (E6.3.3) we obtain te following 0 ( m + m ) dt / T + m R dv / V (E6.3.4) B vb Integrating Eq. (E6.3.4) we ave A va β V B T C B B B B B (E6.3.5)