AP Physics  Heat Engines


 Jocelyn Dawson
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1 A hysis  et Engines et engines operte y onverting het into work. Exmples of het engines ound  ound! Sys the hysis Khun  ll round us. Gsoline engines, jet engines, diesel engines, stem engines, Stirling engines, &t. Just out nything tht urns fuel to generte het is het engine. You ould e onsidered to e lowteh het engine (ut relly nie one, I m sure we ll gree). et is dded to the engine t high temperture. rt of the het is used to generte work nd the rest of the het is sent to some low temperture environment. Fny terminology used for het engines involves things like: high temperture het reservoir (the soure of the input het), low temperture het reservoir (where you send the het you don t use) sometimes this is lled the low temperture het sink. For gret mny het engines, the het sink will e the tmosphere think of gsoline r engine. Sttionry eletri power plnts might use wter from river. A stem ship would use the se, nd so on. Most engines hve yli opertion steps tht re repeted over nd over in order to do the work. A r engine hs pistons tht repetedly move up nd down produing power from the motion of the piston. Effiieny: he het tht is not onverted to work is lled the wste het. All het engines produe wste het. he wste het is sent to the het sink. he first lw tells us tht the engine n never produe more work thn the het tht went into it. After ll, energy hs to e onserved, orret? he rtio of output work to input work/energy is lled effiieny. e W Q out his is written s in W e for the A hysis est. Q his simply sys tht the effiieny is the solute vlue of the rtio of output work to input het. he first lw sys tht the effiieny nnot e greter thn one. But we ve lredy sid tht some of the input het does not eome work. Doesn t this men tht the effiieny hs to e less thn one? Well, so it is. his ws disovered in 1824 y Sdi Crnot (tully NiolsLeonrdSdi Crnot he hd lot of nmes for Frenh physiist). If n engine ouldn t e one hundred perent effiient, wht ws the est you ould do? Crnot set out to find the nswer to tht question. In his pursuit, he invented thermodynmis. 723
2 Crnot imgined the est possile engine, this would e n idel engine it wouldn t hve frition etween its moving prts nd it wouldn t lose het through the wlls of its ylinders, et. An idel engine would hve no hnge in its internl energy. For the engine: U Q W U 0 so Q W he work done y this idel engine is simply equl to the hnge in het, Q. W Q Q C Q is the high temperture nd Q C is the low temperture. We plug this into the effiieny eqution (for like one yle): W done in one yle e Q dded in one yle e W Q lug in vlue for work we figured out: In W Q QC e But Q in is simply Q. Q Q In in Q Q e Q Q is proportionl to so: e C C his is the effiieny of n idel energy. It is the est effiieny tht it is possile to otin. e C he tempertures must e in Kelvins! No rel engine n hieve this effiieny, ut it is possile to ome lose. he eqution is importnt euse it shows us tht the effiieny depends on the two temperture extremes used y the engine. he higher the temperture the engine opertes t nd the lower the temperture of the het reservoir, the higher the effiieny will e. 724
3 olkien & ehnology: A uli Rdio Commentry y Bill mmk J.R.R. olkien's Lord of the Rings enhnts euse it let's us espe into nother world lled "Middle Erth." Yet, odd s this fntsy world is, it rries n importnt messge for our world. olkien pled t the enter of his sg the question of how tehnology fits into our lives. he story ppers to e out the Quest to destroy ring with inredile powers, ut hidden not elow the surfe is ler messge out tehnology. hroughout the Lord of the Rings olkien often hrterizes evil s tehnology. For exmple, one of the mjor villins, Wizrd lled Srumn, lives in ple olkien lls "Isengrd." olkien, who ws n Oxford rofessor of AngloSxon, knew Isengrd ment "iron yrd", wht we might ll n industril prk. Inside tht iron yrd the evil Wizrd Srumn spends his dys uilding mills, hopping down forest, nd lowing things up. e retes system of tunnels nd dms, nd vents for poisonous gses nd fires. olkien writes tht "wheels nd engines nd explosions lwys delight" Surmn nd his followers. he ide of mhines ppers gin when he desries the evil Srumn s hving " mind of metl nd wheels." In ontrst to this evil were the oits. A simple, smll people who hve n grrin eonomy. olkien one wrote to friend "I m in ft oit (in ll ut size). I like grdens, trees nd unmehnized frmlnds; I smoke pipe, nd [I] like good plin food (unrefrigerted)... olkien lived life s opposed s possile to tehnology. During his lifetime he rejeted trins, television nd refrigerted food. e did own r, ut sold it t the eginning of World Wr II. By tht time olkien pereived the dmge rs nd their new rods were doing to the lndspe. e me to think of the internl omustion engine s the gretest evil ever put upon this Erth. is experienes with wr olored his view of tehnologil hnge. e served in the trenhes during World Wr I nd experiened tehnology s fighter plnes, tnks, omings, nd flmethrowers. "By 1918," he one sid, "ll ut one of my lose friends were ded." Smll wonder he disliked the immense power ehind tehnology. In mny wys the gret theme of the Lord of the Rings is tht no one should hve dominion over the world. he Lord of the Rings is n ntiquest, with its gol to destroy universl power forever. erein lies olkien's messge to us, wht mke his Lord of the Rings still ring true tody. e refused to let the mteril world drw the oundries of life, nd though his smll oits he sserted the individul's right nd responsiility to shpe the deisions nd strutures tht determine their life. Diesel engines re more effiient thn gsoline engines euse they operte t higher temperture. It is expensive to uild mhines tht n operte t high tempertures you need ostly metl lloys tht n hndle the temperture, the engine must e stronger euse it is doing more work, so its omponents hve to e eefed up, nd so on. Engine designers hve to lne trdeoffs like tht etween effiieny nd mnufturing expense. A stem engine opertes on wrm 28.0 C dy. he sturted stem opertes t temperture of C. Wht is the idel effiieny for this engine? 725
4 An importnt thing to rememer is tht the tempertures in the eqution must e in Kelvins. hot = C = 373 K ool = 28.0 C = 301 K e h h C 373 K 301 K e 373 K he effiieny n e left s deiml frtion or it n e expressed s perent. o inrese the effiieny of rel engine, one n: Inrese the temperture of the high temperture het reservoir. Derese the het sink temperture. Do oth. he other thing tht n e done is to try nd mke it ehve s lose to n idel engine s possile. his n e done y eliminting frition, deresing the mss of moving prts, &t. No rel engine n hve this effiieny. Eliminting frition nd ll the other good engineering things tht n e done will help the engine pproh the idel effiieny. Unfortuntely it will never reh this level of performne. Wht shme. Digrms: ressure/volume grphs re of tremendous vlue in nlyzing the performne of het engines. Let s look t prt of thermodynmi yle. We will look t the work for eh step in the proess. 1 2 ere is the first step  the proess represents n isori ompression of gs. From to, the pressure remins onstnt its vlue is 1. he volume derese from to 2. his represents work done on the system. It tkes work to ompress the gs. he work is the re under the urve, whih is onveniently shded in so you 2 n see it. he work is: 1. W Now let s look t the next step in the yle. You n see the digrm to the right
5 roess is n isohori ompression. Isohori euse the volume does not hnge ut the pressure inreses. he work for this step is zero. his is euse is zero. W d he next step is proess d. his is n isori expnsion. he gs expnds from to 2, doing work s it expnds. he mount of work is equl to the re under the urve. W he finl step to omplete the yle is proess d. his is n isohori expnsion. he volume stys onstnt, so no work. he pressure dereses. 2 d W he whole yle looks like the grph to the right. he net work done y the mhine in one yle is the re enlosed y the urve. It is the sum of the work, positive or negtive, for eh step in the yle. 2 d he work is just the differene in the two res under the urve tht we hd. W he work done in yle depends on the pth. Let s look t the previous yle nd ompre it with one tht hs different pth
6 2 d Cyle 1 Cyle 2 Cyle 1 is the one we just looked t. he gs is ompressed from 2 to, ompressed from 1 to 2. hen it is expnded from to 2 nd expnded from 2 to 1. retty muh the sme thing hppens in yle 2, exept tht the two expnsions tke ple in only one step insted of two. his differene is signifint however s the net work for yle 1 is greter (twie the mgnitude) thn the net work for yle 2. A het engine s yle is shown in the digrm to the right. 1 = 345 k, 2 = 245 k, 3 = 125 k, nd 4 = 225 k. = 35.0 L nd 2 = 85.0 L. Wht is the net work done during one yle of the engine? We n solve this y finding the re under the urve for the step. hen we find the re under the urve for the d step. he net work is the differene in the two res. Find volume in m 3 : m L 35 x10 m m 1 L m L m 1 L Are for : W A Are Ari Are m m J 1 Ari d 2 728
7 1 Ari m m J W J J J Are for d: Wd Ad Are Ari 3 3 Are m m J 1 Ari Ari m m J W J J J W J J J 6.00 kj Net A sustne undergoes yli proess shown in the grph. A work output ours long pth, work input is required long pth, nd no work is involved in onstnt volume proess. et trnsfer ours during eh proess in the yle. () wht is the work output during proess? () how muh work input is required during proess? () Wht is the net work done during the yle? W A A A () re ri x10 10 m x10 10 m W 1 tm 40 L 4 tm 40 L 1tm 1 L 2 1tm 1 L W 122 x10 J 1.22 x10 J Expnsion proess, so negtive work (tm) (L) () his is positive work. W = re of retngle x10 10 m 2 3 W 1 tm 40 L 40.5 x10 J 4.05 x10 J 1tm 1 L () Wnet W W W 729
8 4 4 3 W 1.22 x10 J x10 J x10 J net he Crnot Cyle: Sdi Crnot envisioned perfet mhine tht would hve the gretest possile effiieny tht it ould possily hve. We ve lredy seen how the eqution for this effiieny ws developed. But wht kind of mhine ould do tht? Well, the mhine tht Crnot me up with is simple piston/ylinder devie. he operting sequene of the thing is lled the Crnot yle. ere is the devie (the drwing ove). he sides nd top of the ylinder re insulted so het nnot flow in or out of the system. he ottom is mde of n idel ondutor so tht het n flow in or out of the system through the ottom of the ylinder. hree stnds re ville for ylinder plement; hot stnd, old stnd, nd perfet insultor stnd. Let s see how the yle works. Step1: Isotherml expnsion. he ylinder is pled on high temperture het sink tht is t. he het is onduted through the ottom of the ylinder nd the gs sors het. We ll this het Q in. As result of the dded het, the gs expnds, pushing the piston upwrd. his step does some work. We ll this step isotherml expnsion euse the temperture stys onstnt (isotherml mens onstnt temperture) nd the volume inreses. he work done is equl to. his step is represented s the urve AB on the digrm elow. Step 2: Aditi expnsion: he ylinder is immeditely moved from the hot stnd to the insulted stnd. One it is pled on the insulted stnd, het n no longer flow into the system (or out of it). he gs ontinues to expnd, ut sine het is no longer entering or leving, this is n diti expnsion. he pressure in the ylinder drops to its lowest vlue. his is represented y the urve BC on the digrm. Work ontinues to e done y the system during this step. Step 3: Isotherml ompression: he ylinder is immeditely ple on the low temperture het sink. et, whih we ll Q out, flows from the ylinder into the het sink. he system loses 730
9 het, the volume dereses, nd the gs is ompressed isothermlly. his is represented y the urve CD on the digrm. Step 4: Aditi ompression: he ylinder is ple on the insulted stnd. et no longer n enter or leve, so the system undergoes diti ompression k to its originl stte long the urve AD. he het engine is now redy to undergo nother yle. ere is the digrm for the Crnot yle. A to B  isotherml expnsion. inreses nd dereses B to C  diti expnsion. C to D  isotherml ompression D to A  diti ompression Work is done long the urves AB nd BC. he effiieny of the Crnot yle, how well it opertes, depends on the sorption of het nd the loss of het in the respetive steps of the yle. One of the key ftors tht ontrols the flow of het is the temperture differene. his would e the temperture of the hot stnd nd the old stnd. When pled on the hot stnd, het flows into the ylinder nd it rehes (if left long enough) the sme temperture s the hot stnd. When pled on the old stnd, the temperture differene is equl to the hot temperture minus the old temperture (of the old stnd). If is inresed, het will flow fster nd the mhine will operte more effiiently. So the higher the hot temperture reservoir (the hot stnd), the greter the mount of het sored y the system. Also, if we derese the old temperture reservoir, this too will inrese the mount of het tht flows. Inrese the het flow nd you inrese the effiieny of the system. 731
10 Der Dotor Siene, You know those little irds filled with red stuff tht o up nd down drinking from glss of wter? I n't mke mine stop, even when I tke wy his wter. e won't stop. lese help!  Joey erry from Springfield, MO. Dr. Siene responds: hose little irds re tully perpetul motion mhines. One you get them strted, they'll never stop  ever  not until the sun winds down nd our glxy goes nov. Even then, somewhere in the imploded lk holes tht ws one our solr system, little irds will e oooing  even in the noiseless vuum of spe. he "red stuff" inside these irds is tully neutrino solution kept in ple y mysterious new fore in the universe lled hyperhrge, whih is ross etween nuler ond nd superglue. Fortuntely, most of these irds re wht siene lls "roken" nd will not o t ll. his is true, t lest for most of us, who ought these little irds t irport gift shops nd we n only thnk the powersthte. Otherwise, they'd just give us the reeps. Boing, oing, oing through eternity, through entropy, oing, oing, oing, oing... Solitude Ell Wheeler Wilox Lugh, nd the world lughs with you; Weep, nd you weep lone. For the sd old erth must orrow its mirth, But hs troule enough of its own. Sing, nd the hills will nswer; Sigh, it is lost on the ir. he ehoes ound to joyful sound, But shrink from voiing re. Fest, nd your hlls re rowded; Fst, nd the world goes y. Sueed nd give, nd it helps you live, But no mn n help you die. here is room in the hlls of plesure For long nd lordly trin, But one y one we must ll file on hrough the nrrow isles of pin. Rejoie, nd men will seek you; Grieve, nd they turn nd go. hey wnt full mesure of ll your plesure, But they do not need your woe. Be gld, nd your friends re mny; Be sd, nd you lose them ll. here re none to deline your netred wine, But lone you must drink life s gll. 732
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