herml & Kineti Physis: Leture Notes Kevin Donovn 4. HA NGINS. 4. ffiieny of Het ngines, Refrigertors n Het Pumps It hs now een estlishe tht in tking flui roun lose yle on P igrm i) he internl energy is unhnge, U = 0 ii) he net work one on/y the system epens on the etils of the pth tken. In lokwise yle net work is one y the system wheres the sme yle ounter lokwise results in net work one on the system. iii) he het flow into/from the system epens on the etils of the pth tken. In lokwise yle net het flows into the system wheres the sme yle ounter lokwise results in net het flow from the system. iv) From the first lw, = - A reversile yle is ny yle tht n e operte reversily, in one sense soring het n oing work while in the other giving out het while work is one on it. For true reversiility the het flow must tke ple with infinitesiml temperture ifferenes etween the system n surrounings. In rel engines whih re not reversile (frition, finite pressure, leky vlves n temperture ifferenes) normlly het is fe in t high temperture n wste het is isre t low (usully mient) temperture. eg. SAM NGIN Superhete stem in. onense wter out PROL NGIN Hot ignite petrol vpour + ir in ooler exhust gses out. As ll engines my e thought of s het engines with hot soure reservoir n ol reservoir for isre het the onept of n engine is ielise to mke it esier to nlyse. he ielistion involves high temperture het reservoir t temperture n low temperture het reservoir t temperture. A working system will extrt het, from the former, eliver het,, to the ltter n o n mount of work,. NB. he het reservoirs y efinition n lose or ept het without their temperture hnging, 86
nvironment herml & Kineti Physis: Leture Notes Kevin Donovn Het Reservoir orking System & yle Het Reservoir Aove is the shemti of the engine, this shemti n similr will e use frequently in the following isussions/esriptions of het engines, refrigertors n het pumps. All three of these evies use working sustne to lter the environment in some wy, y oing work, extrting het or elivering het respetively. i) ffiieny of Het ngine. It is esy to evise meningful mesure of the effiieny or figure of merit for n engine. As its nme suggests the lrger it is the etter. effiieny figure of merit ht is ht wnte out is put in In the se of the engine the esire output is work,. o hieve this het from the hot reservoir,, is put in. It follows from the efinition of effiieny or figure of merit tht this is given y 87
herml & Kineti Physis: Leture Notes Kevin Donovn he first lw my e use to pt this s follows. he system opertes in yle n therefore U 0 ( ) Note the signs! In using the ove eqution the mgnitue of hs een ssume ie. ll vlues re positive. If the sign onvention ws rigorously pplie everywhere then this eqution woul e written; U 0 ( ) is of ourse negtive s it flows out of the system. Stying with the first form of these equtions n unerstning to refer to the mgnitue of the het flow (its iretion eing ssume in the moel). n therefore his question of mgnitue of will e importnt when it omes to lulting tul effiienies. Another wy of looking t this question is to note tht is negtive (it flows from the system) n therefore if the mgnitues weren t use woul e negtive n, sitution giving more thn 00% onversion of het to work n therefore unphysil n impossile!!! 88
nvironment herml & Kineti Physis: Leture Notes Kevin Donovn ii) ffiieny of Refrigertors n Het Pumps. Het Reservoir orking System & yle Het Reservoir Refrigertors re simply het engines yle in the opposite iretion, work in tkes het out s shown in the ove shemti where work,, is one on the working system n het, is extrte from the ol reservoir t the en of yle. In the se of the refrigertor the esire output is het,, eing remove from the ol reservoir (ol shelf in refrigertor). o hieve this work,, is supplie (from power supply) n some of the het remove,, goes to the hot reservoir (the k of the refrigertor or externl environment). It follows from the efinition of effiieny or figure of merit tht this is given y R n therefore 89
herml & Kineti Physis: Leture Notes Kevin Donovn R he het pump is ientil in opertion to the refrigertor exept tht the fous is on elivering het to the hot reservoir. his esirle het, is elivere t the expense of oing work,. An using the first lw it my e written s in HP Out HP Iel (rnot) ngine Si rnot (796 83) ws one of mny who ske, n ttempte to nswer, the very importnt question wht form of engine will give the gretest effiieny? Looking t the effiieny of n engine e nee yle tht will minimise the het rejete to the low temperture reservoir, n mximise the het into the system s extrte from the hot reservoir,.si rnot, who, it is worth relling, egn his seminl work whilst it ws still elieve tht het ws sustne, nmely lori tht ouln t e rete or estroye, me to the onlusion, in 84, tht wht ws require ws reversile yle. his woul e yle in whih ny het flows shoul tke ple with little or no temperture ifferenes. o hieve this he propose yle ompose of n isotherml expnsion of the system t the temperture of the hot reservoir with het trnsfer to the system from the reservoir with no temperture ifferene n n isotherml ompression of this system t the temperture of the ol reservoir with het trnsfer to tht reservoir from the system t 90
herml & Kineti Physis: Leture Notes Kevin Donovn. he two isotherms re joine y two iti proesses (no het flow) tking the expne sustne from the high to the low temperture n tking the ompresse sustne from the low temperture to the high temperture. he rnot yle P Aiti, Isotherm, Aiti, Isotherm, he rnot yle is shown in the igrm ove. he four reversile proesses ssoite with the rnot yle my e summrise; i) Reversile ition of het n expnsion t onstnt temperture ii) Reversile iti expnsion to temperture iii) Reversile rejetion of het n ompression t onstnt temperture iv) Reversile iti ompression k to temperture Before ontinuing to fin the optiml effiieny ttinle with het engine, one running on the rnot yle, rief interlue is neee to look t; 9
herml & Kineti Physis: Leture Notes Kevin Donovn 4. Kelvin-Plnk n lusius Sttements he seon lw originte s n empiril sttement out the limittions of het engines. here re two erly sttements of the seon lw me fter empiril oservtion of how the rel worl ehve; i) he Kelvin-Plnk Sttement: It is impossile to evise evie tht, operting in yle, proues no effet other thn the extrtion of het from single oy ( het reservoir) with the proution of n equivlent mount of work. ii) he lusius Sttement: It is impossile to evise evie tht, operting in yle, proues no effet other thn the trnsfer of het from ooler to hotter oy (het reservoir). Kelvin Reservoir lusius Reservoir > System = System Reservoir IMPOSSIBL Both sttements, while seemingly ressing ifferent spets of the question of het, re in ft equivlent. hey re shown shemtilly in the two igrms ove n the equivlene of the two sttements is emonstrte y onsiering the omposite het engines shown in the igrms elow. 9
herml & Kineti Physis: Leture Notes Kevin Donovn + = R omposite refrigertor < < IMPOSSIBL he igrm on the left ove shows n engine n refrigertor running etween the hot n the ol reservoir with the engine with its yle juste to provie the work tht runs the refrigertor. Only if the Kelvin sttement ws inorret oul the engine in priniple e me to work. If it worke then it oul e use to run the refrigertor etween the hot n ol reservoir. he refrigertor itself oeys the first lw tht is to sy het out equls het plus work in. However the engine plus refrigertor n e onsiere omposite unit n the omposite unit is equivlent to omposite refrigertor extrting het from the low temperture reservoir n elivering it to the high temperture reservoir, het flowing from ol to hot with no input of work, violtion of the lusius sttement. 93
herml & Kineti Physis: Leture Notes Kevin Donovn R = omposite ngine = < < IMPOSSIBL he igrm ove left shows refrigertor tking het from low temperture reservoir n elivering it to high temperture reservoir with no input of work, in ontrvention of the lusius sttement, n n engine, oth operting etween the sme two reservoirs with the engine elivering work n wste het, to the ol reservoir. he engine is oeying the first lw. he left hn igrm n e eonstrute into the right hn igrm. his shows n mount of het extrte from hot reservoir with the elivery of n equivlent mount of work, violtion of the lusius sttement. he two sttements, whih otherwise pper to e ressing very ifferent questions re thus shown to e logilly equivlent. 94
herml & Kineti Physis: Leture Notes Kevin Donovn 4.3 rnot s heorem rnot s theorem simply sttes tht no het engine operting etween two given het reservoirs n e more effiient thn rnot engine operting etween those sme two reservoirs A rnot engine s riefly mentione erlier is; i) A reversile engine ii) An engine operting etween two het reservoirs. o hieve this; het is trnsferre reversily (tht is t zero temperture ifferene) to n from the system on yle of two isotherms t the high n the low temperture linke in yle y two iti urves. his is reversile engine with het trnsfer tking ple t onstnt system temperture ie. isothermlly. Proof. (reutio surum) Suppose suh n engine,, with i exist n i n mount of work,. hoose rnot engine tht oes the sme work. If the rnot engine is reverse it ts s refrigertor. his n e represente on igrms s shown elow. / = = - = - = - Looking t the upper left hn igrm, the effiieny of the engine is 95
herml & Kineti Physis: Leture Notes Kevin Donovn It therefore follows Now turn ttention to the upper right hn igrm. ithin the she ox the rnot engine is eing run in reverse n ting s refrigertor. Following the igrm with its inputs n outputs to the two reservoirs, the omposite system extrts positive het from the ol reservoir n elivers n mount of het to the hot reservoir. But he omposite system is then in isor with the lusius sttement n the premise tht nnot e orret. On the other hn, if then n no net het is elivere. his is llowe so it hs een proven reutio surum tht thus proving the rnot theorem. his proof is vli for ny truly reversile engine s tht ws the only property tht the proof relie upon, the etils of the yle not eing mentione. 96
herml & Kineti Physis: Leture Notes Kevin Donovn A orollry to this theorem is tht ll rnot engines running etween the sme two tempertures hve the sme effiieny. o emonstrte this / / / = / - = - e imgine two rnot engines running etween the sme reservoirs with their yles juste to eliver the sme mount of work. Now reverse one n use the other to run it s refrigertor. It hs just een emonstrte tht to voi ontrition with the lusius sttement it is neessry tht my now e reverse n. he onlusion, reutio surum, is then, tht. he roles of the two rnot engines. In prtiulr, we will see lter tht for norml engine the effiieny n epen on the etils of the gs use i.e. the effiienies my iffer etween n engine with montomi gs n the sme yle with itomi gs. his nnot e the se for the rnot yle thus; he effiieny of rnot engine is inepenent of the working sustne n n epen only on the equilirium properties of the reservoirs. his is non-trivil remrk!! 97
herml & Kineti Physis: Leture Notes Kevin Donovn 4.4 Asolute emperture Sle illim homson (84 907), lter Lor Kelvin, relise tht this provie metho of estlishing n solute temperture sle. He relise tht if he h rnot engine its effiieny is Inepenent of the mteril of the system n epenent only upon the tempertures of the two reservoirs. ith this importnt oservtion he emonstrte tht it ws in priniple possile to efine thermoynmi tempertures,, for the reservoirs y NB. So fr it is only eing stte tht the rtios of the thermoynmi tempertures re equl to the rtio of the het flows. he sle ftor my e fixe y efining P n is then n Asolute hermoynmi emperture foun in priniple y mesuring the effiieny of rnot engine etween reservoir t the unknown temperture n seon t the triple point of wter P. he temperture so foun is inepenent of ny prtiulr mteril. his proposition is emonstrte y onsiertion of two rnot engines running in series s shown elow where the het rejete y the first engine is equl to the het tken in y the seon engine. 98
herml & Kineti Physis: Leture Notes Kevin Donovn 3 = + 3 3 3 3 3 3 3 3 he two engines on the left operte suh s to leve the reservoir t unhnge with engine juste to eposit the sme mount of het into s is tken y engine 3. By Kelvin s efinition of thermoynmi temperture sle; n It follows y iviing the first y the seon tht 3 3 99
herml & Kineti Physis: Leture Notes Kevin Donovn hus, using rnge of rnot engines omplete temperture sle my e efine. It might e suppose (hope!), tht the solute temperture,, is ientil to the iel gs temperture, G n this my e prove y nlysis of the rnot yle. 3 3 Proof. isotherms P G iti iti G he igrm shows the rnot yle with n iel gs s working sustne. As reminer tht the thermoynmi temperture is eing ompre with the iel gs temperture the susript G is inlue with the gs tempertures, G. i) For the power stroke or isotherml expnsion,, G = G n P = nr G U 0 (isotherm) st Lw gives 0 G nrg P nr ln ii) nrg ln Similrly for the exhust stroke or isotherml ompression,, G = G n P = nr G 00
herml & Kineti Physis: Leture Notes Kevin Donovn U 0 (isotherm) st Lw gives 0 P nrg nr ln G nr ln G nrg nrg ln ln G G ln ln iii) Now looking t the itis where G G n therefore G G his ws use in the erlier eqution proving tht G G his is still just equting the rtios of the two tempertures mesure on their prtiulr temperture sle ut with the further requirement tht; it is the se tht P P 73.6 = G (= Kineti = ) Returning to the effiieny of the rnot yle, 0
herml & Kineti Physis: Leture Notes Kevin Donovn 0 An this is the mximum theoretil effiieny of ny engine operting etween those two tempertures. NB. wo importnt points.. he type of gs (montomi, itomi et is irrelevnt s roppe out of onsiertion n. = only if = 0! If the rnot engine is run in reverse it is refrigertor. he effiieny of rnot refrigertor is ) ( rnot R It n lso e operte in reverse s het pump with effiieny; HP rnot ) ( Just quik oservtion, If the reservoir tempertures re very lose then the het pump n refrigertor effiienies eome very lrge, in ft muh greter thn ut the engine effiieny eomes very smll pprohing zero. he sme is true of non-rnot systems. hinking out this it is ler tht refrigertor require to keep its ontents t
herml & Kineti Physis: Leture Notes Kevin Donovn temperture only slightly ooler thn the mient temperture will not e require to work very hr, will e smll n xmple. R will e very lrge. A rnot het pump is esigne to operte etween = 83 K n = 300 K he effiieny is xmple. HP ( rnot ) ( rnot ) R 83 300 300 83 83 300 300 7.6 7 83 6.6 7 7 300 0.057 A stem engine operting with superhete stem t 00 0 n exhust t 0 0. e n give n upper limit of the rnot effiieny. = 473.5K, n = 83.5K 83.5 0.40 473.5 4.5 Rel ngines.. he Otto (Petrol) yle. P Aiti (Power stroke) Aiti xhust 03
herml & Kineti Physis: Leture Notes Kevin Donovn he petrol engine is sprk ignition yle. Petrol is light n voltile oil n vpour is mixe with ir in the piston s ompression tkes ple susequently eing ignite with sprk. he Otto yle is resonle pproximtion to the petrol engine ut with two strokes, expnsion followe y ompression. It is epite ove with the expnsion n ompression unergone itilly (no het input from externl soures). Joining these two iti urves re isohores, (no hnge in volume). his is reily nlyse engine yle s the tools hve een lrey evelope n re to hn. Anlysis is the ignition proess where the fuel is ignite using sprk, het is introue into the system n pressure rpily inreses. It is n isohori proess n no work is one s = 0. he het n e lulte from the first lw 3 U U U nr( ) lerly is positive s n it is therefore flow of het into the gs. ( ) is the power stroke, n iti expnsion where work is one y the gs n the usul eqution hols; is n isohori proess where the gses re exhuste n the system loses het. No work is one s = 0. he het ejete t this stge n e lulte from the first lw n for montomi iel gs; U 3 U U nr( ) lerly is negtive s n it is therefore flow of het out of the gs. ( ) is n iti ompression where work is one on the gs. In typil engine this work is provie y the work of the power stroke prtilly ouple k to hieve the ompression. he ompression rtio is ientil to tht of the power stroke ut the ompression ours t lower pressures n so not ll the work of the power stroke is 04
herml & Kineti Physis: Leture Notes Kevin Donovn 05 require to hieve the ompression n useful work my e extrte. he usul iti nlysis gives; he effiieny is ) ( ) ( he effiieny is usully require in terms of volume rther thn temperture s it is the volume tht is uner the ontrol of engineers n to hieve this the nlysis of the iti proesses is employe. Sutrting the seon iti eqution from the first Defining the ompression rtio r r For ompression rtio of 8 n = 5 7 =.4 for rigi itomi gs. 0.54
herml & Kineti Physis: Leture Notes Kevin Donovn. he Diesel yle. P Aiti (Power stroke) Aiti (Ignition stroke) xhust Unlike the petrol engine se on sprk ignition the si esign of iesel engine uses the phenomenon of ompression ignition; s ir is ompresse it gets hotter n t high enough temperture, with little it of fuel mixe in, it ignites riving the susequent power stroke expnsion. Diesel oil unlike petrol is not voltile n this requires tht the liqui fuel is isperse s spry into the piston. he iesel engine yle is shown ove. It my e nlyse s usul to fin n n thus the effiieny. nlysis is n isori expnsion use y ignition of fuel with the sorption of het. t onstnt pressure y the system. P is onstnt so P n; P P ( ) here will lso e smll mount of inientl work one y the system; 06
herml & Kineti Physis: Leture Notes Kevin Donovn 07 P P P is n iti expnsion whih is the power stroke of the yle with work eing one y the engine. Anlysing this s n iti proess; is n isohori proess, llowing exhust gses to e relese with pressure rop to the yles lowest pressure (typilly to tmospheri pressure) n het is rejete t onstnt volume y the system in this exhust. = 0 so = 0 From the first lw (ssuming montomi iel gs). ) ( ) ( 3 nr U U U is het flow out s it flows from higher to lower temperture n the sign is negtive. herefore the solute vlue of must e tken for when use in the effiieny equtions. ( ) is n iti ompression is the ignition yle. If ir is suenly ompresse it ignites n with little it of injete fuel the ignition provies enough het to rive self sustining yle. For this proess; It is lso known tht nr P nr P n P P he ingreients re now ssemle to onlue the nlysis;
herml & Kineti Physis: Leture Notes Kevin Donovn 08 he effiieny n now e written s; ) ( P he effiieny is require in terms of volumes n to hieve this; Using An similrly he effiieny n now e written in terms of volumes only s nels in the quotient; Defining engine prmeters; r is the expnsion rtio r is the ompression rtio An refully seleting the next step y multiplying top n ottom y
herml & Kineti Physis: Leture Notes Kevin Donovn 09 r r r r
herml & Kineti Physis: Leture Notes Kevin Donovn 3. he Stirling yle. he Stirling yle is n exmple of nother engine yle. he Stirling engine opertes on lose system n offers quiet performne with potentilly high effiienies. It s tion my e esrie s follows; i) Reversile ition of het n expnsion t onstnt temperture = ii) iii) Het rejetion n ooling t onstnt volume from to Reversile rejetion of het n ompression t onstnt temperture = iv) Aition of het t onstnt volume k from to temperture P 4 Isotherm t 3 Isotherm t It hs een note lrey tht s oppose to the rnot yle, here s well s the het flow isothermlly there is lso het e n rejete uring the isohori proesses n s het is e with no work one this shoul reue the effiieny of the Stirling yle when ompre with the rnot yle. Further, tht het is e or extrte irreversily n the yle is therefore irreversile. 0
herml & Kineti Physis: Leture Notes Kevin Donovn o otin the effiieny of the Stirling het engine ifferent pproh is tken where net work one, Net, n e nee to e foun over the yle n Net. e ork is only performe on or y the system uring the isotherml expnsion n the isotherml ompression no work eing performe on the isohores. nr ln nr ln Net nr ln NB. his is net work whih is negtive s the system oes net work on its environment s n e seen from the ft tht n tht he het e to the system ours oth uring the isotherml expnsion n the isohori temperture rise, proess Isotherml expnsion ; U 0 Isohori temperture rise nr ln 4 herefore the het e uring the yle is; e nr ln
herml & Kineti Physis: Leture Notes Kevin Donovn Putting this together for the effiieny; nr nr ln ln e S NB. In the ove eqution the solute mgnitue hs een use for the work (ie it is me positive y inverting the orer of the volumes in the logrithm of the numertor). Divie top n ottom y nrln nr ln S An then ivie top n ottom y nr ln S his gives us the Stirling effiieny in terms of the rnot effiieny; nr nr ln ln S From this result it is immeitely seen tht the effiieny of Stirling engine is lower thn the effiieny of rnot engine running etween the sme two reservoirs.
herml & Kineti Physis: Leture Notes Kevin Donovn Further, relling tht for n iel gs U s nr where s is the numer of egrees of freeom ( 3 for montomi gs, 5 or 7 for itomi moleule epening on whether it is rigi or virtes respetively) S s ln NB. he Stirling engine will perform etter for montomi gs. he rel worl. It is time to signl note of ution fter the foregoing nlysis. Four yles hve een nlyse, the iel rnot yle, the Otto, Diesel n Stirling yle suggesting tht the ltter three re more like rel engines. In mny wys of ourse they re exept for one ig ifferene. Any rel engine opertes with losses or issiption tht hve not een inlue here in this nlysis, sometimes etween more thn two het reservoirs, with leky vlves n with frition in the moving prts n the Otto, Diesel n Stirling yles s nlyse here were themselves ielistions, tht is, s pproximtions to rel yles. lerly the nlysis of rel engine through thermoynmis is iffiult n involve tsk. ht hs een presente here is wy of pprohing the tsks y reking the jo own into mngele piees. It hs lso een very useful prtie t thermoynmi nlysis. Rel refrigertors, gin operte on irreversile yles n re iffiult to nlyse theoretilly. hey use working sustne, the refrigernt, tht must e vpour when t the operting temperture of the ol reservoir otherwise the operting meium woul onense. his is the reson for mny of the exoti refrigernts tht hve een use over the yers. Refrigernt hemil BP 0 F BP 0 Ammoni NH 3-33.35 Freon ` l F -.6-9.8 DihloroDiFluoroMethne l F 4 38 3.3 l 3 F 75.3 4. l 3 F 3 8 47.8 3
herml & Kineti Physis: Leture Notes Kevin Donovn hlorofluoro rons, Fs hve ominte ut re now reognise s environmentlly uneptle. In the 9 th entury mmoni eme the refrigernt of hoie llowing perishles suh s fish to e ught in Newfounln on the grn nks n then trnsporte roun the worl reting the o inustry. Ammonis is retking its position s top refrigernt one gin euse of its enign environmentl impt on the Ozone lyer n s n gent of glol wrming, the Fs hving long sine een none. Refrigertion is usully hieve y expnsion of the sturte liqui through vlve, the temperture n pressure eing oth lowere signifintly in the proess, with up to 0 tm rop in pressure prouing mix of liqui n vpour in o-existene. For this to work using the Joule Kelvin effet positive Joule Kelvin oeffiient is neessry. it is to e relle tht there is n inversion temperture elow whih the oeffiient will e positive n n expnsion will use rise in temperture. lerly the gs nnot e use in suh refrigertion yle ove the inversion temperture. 4