ESCI 341 Atmospheric Thermodynamics Lesson 11 The Second Law of Thermodynamics

Transcription

1 ESCI 341 Atmosperi ermodynamis Lesson 11 e Seond Law of ermodynamis Referenes: Pysial Cemistry (4 t edition), Levine ermodynamis and an Introdution to ermostatistis, Callen HE SECOND LAW OF HERMODYNAMICS ere are two ommon ways of stating te Seond Law of ermodynamis. Statement #1: In an isolated system, entropy never dereases ( Statement #2: It is impossible for a system to undergo a yli proess wose S isol 0 sole effets are te flow of eat into te system from a eat reservoir and te performane of an equivalent amount of work by te system on te surroundings. Statement #2 is alled te Kelvin-Plank statement, and asserts tat you annot onvert a given amount of eat into an equal amount of work (tere is no su ting as a 100% effiient engine.) Even toug te two statement look very different, tey are atually equivalent! ). PROOF HA SAEMEN #1 AND SAEMEN #2 ARE EUIVALEN Imagine te yli proess sown on te termodynami diagram below, were β and η are arbitrary termodynami variables. Leg 1: A reversible, isotermal proess from State 1 to State 2. e ange in entropy of tis proess is S1 rev (1)

2 Leg 2: A reversible, adiabati proess to State 3 (tere is no ange in entropy during tis proess). Leg 3: An irreversible, adiabati proess bak to State 1. e entropy ange during tis leg we will all S3. Sine entropy is a state variable, and sine we end up at te same point as we started, te total entropy ange of te yle must be zero. erefore, we ave S3 S1 rev. (2) We also know tat, sine internal energy is a state variable, ten U rev W 0. (3) is means tat te work done in te yli proess is equal in magnitude, but of opposite sign, to te eat added to te system, W rev. (4) If rev were positive (meaning eat is added to te system), te work would be negative. is means te system would do work on te surroundings. e net result of tis would be a yli proess in wi all te eat added to te system is onverted into work, wi violates te Plank statement. erefore, rev must be negative. A negative rev doesn t violate te Plank statement, sine in tis ase work is being done on te system, and eat is being extrated. is is perfetly allowed by te Plank Statement, as work an be onverted to eat wit 100% effiieny. e Plank statement says tat you an t onvert eat to work wit 100% effiieny, but you are allowed to onvert work to eat wit 100% effiieny. We ve proven tat rev 0. is means, from Eq. (2), tat S. (5) 3 rev 0 us, te entropy must inrease in an irreversible, adiabati proess. If a system is isolated, it is adiabati. erefore, in an isolated system, any irreversible proess will result in an inrease in te total entropy of te system. Purely reversible proesses in an isolated system will leave te entropy unanged. 2

3 e entire universe is an isolated (and losed) system (at least as far as we know). erefore, we an also write universe S 0. We ve just proven tat te matematial statement, S isol 0, and te Kelvin- Plank statement, are equivalent! FURHER COMMENS ON HE SECOND LAW Some tings to remember: In any irreversible, adiabati proess, entropy inreases. is is true, even if te proess is quasi-stati. In any reversible, adiabati proess, entropy remains onstant. In an isolated system, reversible proesses don t ange te total entropy of te system. In an isolated system, irreversible proesses always inrease te total entropy of te system. e entropy in an isolated system an never derease. It an only remain onstant (if te proesses are purely reversible) or inrease (if tere are any irreversible proesses). e universe is an isolated system (so we believe): Sine tere are ertainly lots of irreversible proesses at play in our universe, ten te total entropy of te universe is always inreasing. e seond law of termodynamis is a big downer! It says we will never be able to onstrut a 100% effiient eat engine, or ave a perpetual motion maine. Pilosopial note: e seond law of termodynamis, like te first law, as never been proven to be true. In fat, it annot be proven to be true. It is a priniple or law tat so far as always worked. But, tat s not to say tat at some point in time a proess or penomenon will be disovered tat violates te seond law of termodynamis. In tat event, ten te seond law will be proven to be false. 3

4 HE MAXIMUM WORK HEOREM e seond law of termodynamis tells us tat tere is no su ting as a 100% effiient eat engine (one tat an onsume a given amount of termal energy and onvert it ompletely into work). We an use te seond law of termodynamis to investigate te maximum teoretial effiieny of a eat engine. is is not just an abstrat idea for meteorologists. Many meteorologial penomenon, su as tropial ylones, beave like eat engines, onverting termal energy into work (kineti energy). We an terefore speak of te maximum teoretial effiienies of tropial ylones, te Hadley irulation, et. Imagine an isolated termodynami system tat onsists of tree subsystems: e primary system: is subsystem undergoes a proess between two termodynami states (State A and State B) and makes available an amount of energy du. e reversible work soure: is subsystem eiter performs work, or as work performed on it. e reversible eat reservoir: is subsystem is a repository for any residual eat. 4

5 From te first law of termodynamis we ave du d dw. e total ange in entropy of te entire system is te sum of te entropy anges of ea individual subsystem, wi is ds total ds primary d were r is te temperature of te reversible eat reservoir. Eliminating d from te two expressions above yields dw du ds ds. r primary In te expression above, te following are fixed: du, dsprimary, and. In order to aieve maximum effiieny, we want to maximize dw. o do tis, we need to minimize dstotal. Ideally, we want to make dstotal = 0, wi would our if te system operates ompletely reversibly. We ave just proven wat is known as te Maximum Work eorem: For all proesses leading from a speified initial state to a speified final state of te primary system, te delivery of work is maximum (and te delivery of eat is minimum) for a reversible proess. Furtermore, te delivery of work (and of eat) is idential for every reversible proess. 1 r r total MAXIMUM EFFICIENCY OF A HEA ENGINE We an use te maximum work teorem to determine te maximum teoretial effiieny of a eat engine. You migt ask wy we would be interested in studying eat engines in an atmosperi termodynamis lass. e answer is simple many meteorologial proess (tunderstorms, urrianes, extratropial ylones, et.) at as eat engines, onverting termal energy into kineti energy (or work). A eat engine is an engine tat takes eat from a ot reservoir, onverts part of tis eat into work, and deposits te remainder into a old reservoir. We will imagine tat te ot and old reservoirs are so large tat teir temperatures remain pratially onstant. 1 ermodynamis and an Introdution to ermostatistis, Callen, 1985, Jon Wiley & Sons 5

6 From te first law of termodynamis (onservation of energy) we ave W. e termodynami effiieny of te engine is defined as W/, and is e ange in entropy is W 1. S. Sine we know te most effiient engine possible is a ompletely reversible engine (for wi S = 0), we an write. Substituting te above expression into te equation for effiieny gives 1. Maximum teoretial effiieny of a eat engine e greater te temperature differenes between te ot and old reservoirs, te greater te effiieny. No real engine an ever be more effiient tan a reversible engine. is means tat te maximum effiieny of any eat engine is real engine 1. 6

7 HE CARNO CYCLE e maximum work teorem, and te maximum effiieny of a eat engine were bot derived witout using any one speifi reversible yle (sine it turns out tat all reversible yles ave te same effiieny). However, most termodynamis texts and disussions make referene to a speifi reversible yle known as te Carnot yle, and istorially te maximum work teorem and even te onept of entropy were developed using te Carnot yle. Beause of its istorial signifiane we briefly disuss tis yle ere. e Carnot yle onsists of four legs: A ig-temperature isotermal expansion, wi requires an amount of eat to be added to te gas. An adiabati expansion. A low-temperature isotermal ontration, wi requires te removal of eat. An adiabati ompression bak to te original starting point. On a P-V diagram te Carnot yle looks like APPLICAIONS OF HEA-ENGINE EFFICIENCY O HE AMOSPHERE At tis point you may be asking Wat in te ek does te effiieny of a eat engine ave to do wit atmosperi termodynamis? is is ertainly a valid question. But it turns out tat studying eat-engine effiieny is very relevant to atmosperi proesses. 7

8 ropial ylones, te Hadley irulation, monsoons, et. an all be tougt of as eat engines tat operate between a ig temperature reservoir (te ropial oeans) and a low temperature reservoir (te upper tropospere), and onvert te eat into kineti energy. A knowledge of te effiieny of tese eat engines an allow us to plae upper limits on te amount of kineti energy tat an be produed. However, atmosperi eat engines, like oter real-life eat engines, are not reversible. o extend te onept of eat engines to real-life situations requires us to use te onept of nonequilibrium termodynamis, or at least finite-time termodynamis. Altoug tese are beyond te sope of tis ourse, a few good referenes or artiles to read on tis subjet are Effiieny of a Carnot engine at maximum power output, F.L. Curzon and B. Alborn, Amerian Journal of Pysis, 43, pp , 1975 Wind energy as a solar-driven eat engine: A termodynami approa, J.M. Gordon and Y. Zarmi, Amerian Journal of Pysis, 57, pp , 1989 A nonendoreversible model for wind energy as a solar-driven eat engine, Journal of Applied Pysis, 80, pp , 1996 Understanding Non-equilibrium ermodynamis, G. Lebon, D. Jou, and J. Casas-Vazquez, Springer-Verlag, 325 pp., 2010 POENIAL EMPERAURE Potential temperature is defined as te temperature tat an air parel would ave if it were brougt adiabatially and reversibly to some referene pressure (almost always 1000 mb). From te Poisson relation involving and p, we ave p 1 p R p onst. so tat potential temperature an be written as R p p0. p 8

9 Potential temperature is onserved during a reversible adiabati proess. erefore, it an be used as a proxy for entropy. In fat, te two are related via s s 0 p ln. 0 Meteorologists often refer to adiabats as isentropes. Isentropi analysis is analysis done on surfaes of onstant potential temperature. EXERCISES 1. An engine operates on a Carnot yle. e working fluid is Helium, and ideal gas wit a moleular weigt of 4 g/mol. e initial pressure and speifi volume are 1000 mb and 6 m 3 /kg. a. Wat is te initial temperature? b. e volume at point B is 5 times greater tan at point A. Wat is te pressure at point B?. e volume at point C is 5 times greater tan at point B. Wat is te pressure at point C? d. Wat is te temperature at point C? 9

10 e. Wat is te pressure and speifi volume at point D? f. Wat is te eat per unit mass added to te working fluid during te ig temperature expansion (A to B)? g. Wat is te eat per unit mass removed from te working fluid during te low temperature ompression (C to D)?. Wat is te speifi work (work per unit mass) done by te working fluid during one yle? i. Using your answers to f., g., and., find te effiieny of tis engine and ompare it to te maximum teoretial effiieny given by Sow tat te maximum teoretial effiieny of a refrigerator (defined as te amount of eat removed from te old reservoir divided by te work) is. 3. Sow tat speifi entropy is related to potential temperature via s s0 p ln. 0 (Hint: Begin wit te expression for s(,p) and use te definition of potential temperature.) 4. a. Explain wy, for an ideal gas, te yli proess sown below on a p-v diagram an only go ounterlokwise or else it violates te Seond Law of ermodynamis. 10

11 b. For some substanes te proess sown migt be able to proeed lokwise. Explain wy tis migt be possible. 11