Classical Thermodynamics 1/e

Transcription

1 4 xt from th Manucrt for Clacal hrmodynamc / Subrata Bhattacharj HIS MAERIAL MAY NO BE DUPLICAED IN ANY FORM AND IS PROECED UNDER ALL COPYRIGH LAWS AS HEY CURRENLY EXIS. No art of th matral may b rroducd, n any form or by any man, wthout rmon n wrtng from th Publhr. 4 Paron Educaton, Inc. Prntc Hall Ur Saddl Rvr, NJ 7458 h ublcaton rotctd by Untd Stat coyrght law, and dgnd xcluvly to at ntructor n tachng thr cour. It hould not b mad avalabl to tudnt, or to anyon xct th authorzd ntructor to whom t wa rovdd by th ublhr, and hould not b old by anyon undr any crcumtanc. Publcaton or wdrad dmnaton (.. dmnaton of mor than xtrmly lmtd xtract wthn th claroom ttng) of any art of th matral (uch a by otng on th World Wd Wb) not authorzd, and any uch dmnaton wll volat th Untd Stat coyrght law. In condraton of th author, your collagu who do not want thr tudnt to hav acc to th matral, and th ublhr, la rct th rtrcton 4-

2 COMPREHENSIVE ANALYSIS OF SEADY SYSEMS In chatr-, th fundamntal law of thrmodynamc- th conrvaton of ma rncl, th frt law, and th cond law wr xrd a balanc quaton of ma, nrgy, and ntroy for a gnrc on untady ytm. Clod tady ytm wr thn analyzd a a cal ca by mlfyng th balanc quaton. Analy of any othr ty of ytm rqur valuaton of tat. Havng tablhd dffrnt matral modl n chatr 3 for valuatng xtndd tat, w ar now n a oton to tackl mor comlx ytm. h chatr ddcatd to th analy of on tady ytm ytm wth all ty of ntracton wth th urroundng yt no chang n thr global tat o that total ma, tord nrgy, or ntroy do not chang wth tm. Examl of uch ytm nclud, nozzl, dffur, um, comror, turbn, throttlng valv, hat xchangr, mxng chambr, tc., oratng at tady tat. h objctv of th chatr to gan nght nto th tady-tat oraton of th dvc through comrhnv ma, nrgy, and ntroy analy. h framwork dvlod for th analy clafy a ytm through utabl aumton, cutomz th balanc quaton, lct an arorat matral modl for th workng ubtanc, mak ncary aroxmaton, obtan a manual oluton, u arorat ES damon to vrfy manual oluton, and carry out what-f tud whnvr obl wll bcom a tmlat for ytm analy throughout th book. Chatr 4-

3 4 COMPREHENSIVE ANALYSIS OF SEADY SYSEMS COMPREHENSIVE ANALYSIS OF SEADY SYSEMS Govrnng Equaton and Dvc Effcnc ES and th On Stady Damon Enrgtc Effcncy Intrnally Rvrbl Sytm Introc Effcncy Comrhnv Analy P, Duct or ub Nozzl and Dffur urbn Comror, Fan and Pum hrottlng Valv Hat Exchangr ES and th Mult-Flow Non-Mxng Damon Mxng Chambr and Sarator ES and th Mult-Flow Mxng Damon Clour Indx

4 4. Govrnng Equaton and Dvc Effcnc Mot ngnrng dvc ar dgnd to orat ovr a long rod of tm wthout much chang n th oratng condton. Ga turbn, tam owr lant, rfrgraton ytm, ndutral ar-condtonng ytm, tc., orat omtm for month bfor chduld mantnanc hut down. Comonnt uch a, turbn, um, comror, nozzl, dffur, hat xchangr, tc., ar on, allowng ma tranfr acro thr boundar, and ar aumd to orat at tady tat o that th nahot takn wth a tat camra ( c..3.) do not chang wth tm undr th dgn condton. On dvc, oratng at tady tat wll b rfrrd a on tady ytm n th book. Condr a artcular on tady ytm, a tam turbn llutratd n Anm..C.turbn and Fg. 4., whch roduc haft work at th xn of flow nrgy a tam a through an altrnatng r of tatonary and rotatng blad. h fxd blad attachd to th tatonary cang of th turbn crat nozzl had aag, through whch tam xand to a lowr rur and hghr vlocty (w wll larn mor about why a nozzl acclrat a flow hortly) bfor mngng on an array of blad attachd to a cntral haft. ranfr of momntum from tam to th blad crat a torqu on th haft, makng t turn. Stam thn ntr th nxt tag and th roc ratd untl t lav th turbn at a tmratur and rur much lowr than tho at th nlt. h larg varaton of rort acro th dvc man that th turbn a non-unform ytm. Howvr, rort at a gvn ont do not chang wth tm at tady tat th global tat, whch an aggrgat of all th local tat, rman frozn n tm. h concluon about th tady turbn hold tru for any on tady ytm, a gnrc vron of whch rrntd by Fg. 4. (or Anm. 4.A.gnrcSytm). h ytm non-unform a ndcatd by th varaton of color, but tady, whch ndcatd n th anmaton by th fact that th color attrn do not chang wth tm. h xtrnal work tranfr n mot ytm cont rmarly of lctrcal and haft work and hat tranfr occur motly wth th atmohrc rrvor (ER) unl ndcatd othrw. Wth th ytm mag rmanng frozn, all global rort of th on ytm, ncludng m, E, and S, mut rman contant at tady tat. Aftr all, th global rort ar um of th corrondng rort of th local ytm comrng th ytm. hrfor, th untady trm n th balanc quaton th tm drvatv of 4-4 Fg. 4. Stam a through a r of nozzl connctd to th cang and blad attachd to th rotatng haft n a turbn. ṁ W xt Q ṁ Fg.4. h gnrc on tady ytm th global tat non-unform but frozn n tm.

5 ma, nrgy, and ntroy can b t to zro, mlfyng th govrnng balanc quaton condrably. Furthrmor, a larg cla of dvc ha only a ngl nlt and a ngl xt, that, a ngl flow through th ytm. For uch ngl-flow dvc, th govrnng quaton ar furthr mlfd a th ummaton of tranort trm ovr multl ort bcom unncary. h rultng quaton ar rroducd blow from chatr and llutratd n Anm. 4.A.govEqn. AV AV Ma: m = m ; or, ρ AV = ρ AV ; or = (4.) v v Enrgy: de dt ( ) = m j j + Q W xt; whr, j h + k + (4.) Entroy: ds dt Q = m ( ) + + S (4.3) gn B Dffrnt trm of th nrgy and ntroy quaton ar llutratd n th nrgy and ntroy dagram of Anm. 4.A.gnrcSytm. For many dvc, chang n kntc and otntal nrg ar oftn nglgbl, allowng th flow nrgy j to b rlacd by nthaly h. Anothr mlfcaton rult n th ca of adabatc ytm a Q t to zro. Not that by ttng th tranort trm to zro (no ma tranfr), th quaton rduc to th t of quaton ud n th analy of clod tady ytm n c ES and th On Stady Damon h on tady damon buld uon th tat damon, ntroducd n cton.3.3, and wll b xtnvly ud n th chatr to vrfy manual oluton and uru what-f tud. On tady damon for ngl flow dvc ar locatd n Damon> Sytm> On> Stady> Gnrc> SnglFlow ag. hr, you wll th famlar matral modl ltd jut a n tat damon ag. Clck on a artcular modl, ay, SL-modl, to launch a ngl-flow damon whr th workng flud a lqud (or old y, t obl to hav an on ytm wth a old ang through t). An on tady damon look rmarkably mlar to th corrondng flow tat damon. Both th damon hav th am dfault vw wth dntcal global control anl and tat anl. wo nw anl dvc and xrgy anl that ar accbl through th tab ar addd to th on tady damon. In th tat anl, you calculat thrmoflud.nt 4-5

6 th nlt and xt tat calld th anchor tat of a dvc a comltly a obl from th gvn nformaton. Now wtch to th dvc anl. In th local control anl, you wll a dvc dntfcaton choc (Dvc-A, B, tc.), and choc for lctng th anchor tat. Undr th control anl you wll vral dvc varabl Qdot, Wdot_xt, tc. Wth th anchor tat calculatd, all you do n th anl to lct a dvc dntfcaton (A, B, tc.,), choo th nlt and xt tat from th calculatd tat tack, ntr th known dvc varabl, and Calculat. h govrnng quaton ar olvd to roduc th unknown. If an unknown han to b a tat rorty (j,, or mdot, for ntanc), t otd back to th arorat tat. You can go back to that tat (by lctng th tat tab and thn th tat numbr) and calculat th tat comltly wth th hl of th nwly valuatd rorty. Altrnatvly, th Sur-Calculat button do th am by tratng btwn th dvc and tat anl. It alo gnrat th ES-cod n th I/O anl that ummarz th oluton, and from whch th oluton can b rroducd at a latr tm. ES-cod for all th xaml n th chatr can b found n th ES.Examl ag. A dcuon of th xrgy anl wll b otond untl chatr 6. For mor dtal and hand-on xaml on th on tady damon, go through th utoral.damon.chatr4 ag or watch th vdo cl n chatr 4 of th vdoour lnkd from th tak bar. 4.. Enrgtc Effcncy Effcncy man varou thng for varou ytm. Howvr, t can b looly dfnd a th rato of drd outut to rqurd nut. In cton., th nrgtc ffcncy for clod ytm wa ntroducd a th rato of drd nrgy to th rqurd nrgy nut for a clod tady dvc. h am dfnton can b xtndd for an on tady dvc. h nrgtc ffcncy, alo known a th frt law ffcncy, for a tady dvc, on or clod, dfnd a Drd Enrgy Outut ηi (4.4) Rqurd Inut of Enrgy whr η rrnt ffcncy of dffrnt ty and th ubcrt I tand for th frt law. Scfc quantt that aar n th numrator and dnomnator dnd on th uro of a gvn dvc and how ach trm n t nrgy balanc quaton ntrrtd hycally. Wth th untady trm drong out du to tady-tat condton, th nrgy quaton for a gnrc dvc, Eq. (.8), can b ntrrtd a follow. 4-6 J W xt Q J Fg. 4.3 Enrgy flow dagram for an on tady dvc hown n Fg. 4..

7 = m j m j + Q W J nt, Nt rat of tranort of nrgy nto th ytm. Nt Rat of hat tranfr nto th ytm. xt ; or, nt Nt Rat of xtrnal work tranfr out of th ytm. J + Q = W (4.5) xt For a tady dvc, th nt nrgy tranortd by th flow nto th dvc lu th nrgy tranfrrd by hat mut qual th xtrnal work dlvrd by th ytm. h ntrrtaton alo vdnt from th nrgy flow dagram of Fg. 4.3 (alo clck th Enrgy button n Anm. 4.A.gnrcSytm). Such dagram can b hlful to dfn and vualz th nrgtc ffcncy for a gvn dvc. W hav alrady calculatd th ffcncy of lctrcal ac hatr a clod tady ytm - to b % n Ex. -9. Lt u rform a mlar analy on an lctrcally owrd watr hatr ( Anm. 4.A.watrHatr) n th followng xaml. EXAMPLE 4- [me ] Analy of a Watr Hatr An lctrc watr hatr ul hot watr at 5 kpa, 7 o C at a flow rat of L/mn a hown n th accomanyng dagram. h watr tmratur at th nlt 5 o C. Du to oor nulaton, hat lot at a rat of kw to th urroundng atmohr at 5 o C. Ung th SL modl for watr and nglctng kntc and otntal nrgy tranort, dtrmn (a) th lctrcal owr conumton, (b) th nrgtc ffcncy, and (c) th rat of ntroy gnraton n th hatr unvr. Aum no rur lo n th ytm. SOLUION Analyz th hatr unvr, nclod wthn th rd boundary of Fg. 4.4, ung th ma, nrgy, and ntroy balanc quaton. Aumton h hatr at tady tat. Flow tat at th nlt and xt, tat and, ar unform and at LE. Snc k and ar nglgbl, j h + k + h. Analy h matral rort for watr, ρ and c v, ar obtand from abl A- or th SL tat damon a 997 /m 3 and 4.84 / K rctvly. h ma flow rat can b obtand from th volum flow rat by ung Eq. (3.4). 3 m = ρv = 997 =.66 6 J ( W l ) ( Q) J Fg. 4.4 Sytm chmatc and nrgy flow dagram for Ex. 4- (vt Anm. 4.A.watrHatr for th ntroy dagram). Analy nvolvng ma, nrgy, and ntroy quaton ymbolzd by mee n brackt. 4-7

8 h nrgy quaton, Eq. (4.), can b manulatd to roduc th lctrcty conumton, W l = W, a follow. xt ( ) xt ( ) ( ) = m j j + Q W W = m j j + Q m h h + Q xt mc v ( ) mv ( ) ( )( )( ) ( ) = + + Q = = 4. kw In th drvaton, th formula for nthaly dffrnc ubttutd from Eq. (3.4). h ngatv gn ndcat that xtrnal work dlvrd to th hatr. h xtrnal lctrcal work clarly th rqurd nut whl th nrgy m j j, th drd outut. h nrgtc ffcncy, uld to th watr, ( ) thrfor, can b xrd a m ( j j ) Q W xt ( ) ( 4.) 38. ηi = = = = = 95. % W W xt xt ( ) For th ytm unvr nclod wthn th rd boundary of Fg. 4.3, th ntroy quaton, Eq. (4.3), roduc ds dt Q = m + + S ( ) gn,unv Q Q ( ) Sgn,unv = m = mcv ln ( ) ( ) ( ) ( 73 ) kw = (.66)( 4.84) ln = K ES Analy Launch th on tady SL damon and lct Watr(L) from th workng-flud mnu. Evaluat tat- from,, and Voldot, and tat- from =,, and mdot=mdot. In th dvc anl, lct tat- and tat- a th nlt and xt tat, ntr Qdot a - kw, and t _B a 5 dg-c. Calculat. Wdot_xt and Sdot_gn ar valuatd, vrfyng th manual rult. By changng to any othr valu, and urcalculatng, th rult can b hown to b ndndnt of th hatr rur. 4-8

9 Dcuon Entroy gnratd nd th hatr a lctrcal nrgy datd nto hat (lctronc frcton) and a hat tranfrrd acro a fnt tmratur dffrnc (thrmal frcton) nd and n th mmdat urroundng of th ytm. An nrgtc ffcncy of 95% may m qut atfactory. Howvr, a w wll larn n chatr 6, th accomanyng ntroy gnraton a maur of dgradaton or watfuln of uful nrgy. h am goal of rang th watr tmratur could b achvd wth much l work f a rvrbl watr hatr could b nvntd Intrnally Rvrbl Sytm Although ntroy gnraton rvav n mot ytm, dalzd ytm, whr ntroy gnraton can b nglctd wthn th ytm boundary or vn n th ntr ytm unvr, rv a mortant bnchmark for ral ytm. Such ytm hav bn ntroducd n c...3. a ntrnally rvrbl and rvrbl ytm rctvly. Bcau ntroy gnraton can b lookd uon a gnralzd frcton, an ntrnally rvrbl ytm oftn form an dal countrart of an actual dvc. Analy of uch dalzd ytm ( Anm..D.rvrblty), can rovd gnfcant nght nto undrtandng th rformanc lmt of many dvc. o xlor th mlcaton of rvrblty n an on tady ytm, lt u do an ntroy analy of a gnrc ntrnally rvrbl ( S gn,nt = ) ngl-flow ytm, ktchd n Fg h uffx nt.rv. addd to both hat and xtrnal work tranfr to rmnd u that th ytm ntrnally rvrbl. Bcau thr no ntrnal thrmal frcton, tmratur mut vary n a contnuou mannr from th nlt to th xt. h chang n ntrnal boundary tmratur rqur th ntroy tranfr by hat to b rrntd by an ntgral, and th ntroy quaton can b manulatd a follow. ds dt = + + dq nt.rv. m ( ) ds gn,nt = = dq = md dq nt.rv. dqnt.rv. m d ; md nt.rv. (4.6) For th ngl-flow ytm, Eq. (4.6) can b ntgratd from th nlt to th xt, yldng Q nt.rv. local ntrnal ytm dw xt dq nt.rv. Fg. 4.5 Locaton of tat and ar o clo togthr that th local ytm (dottd boundary) can b condrd unform. W xt,nt.rv. 4-9

10 nt. rv. m d kw [ ] Q = (4.7) Although drvd on urly thortcal ground, th quaton ha an mortant mlcaton for any dagram. No mattr what th workng flud, th ara undr th dagram ( Fg. 4.6) can b ntrrtd a th hat tranfr r unt ma of th workng flud a long a th ytm can b aumd to b ntrnally rvrbl. Of cour, th am ntrrtaton alo al to rvrbl ytm ( S gn,unv = ), whch mut b ntrnally rvrbl. Hat tranfr to an ntrnally rvrbl ytm calld rvrbl hat tranfr rrntd by Eq. (4.7). h gnfcanc of rvrbl hat tranfr can b arcatd whn w dcu rvrbl cycl n chatr 7 through. For rrvrbl ytm ( S gn > ), t can b hown from Eq. (4.6) that th ara undr th dagram can hav contrbuton from hat tranfr a wll a ntrnal rrvrblt rrntd by th ntrnal ntroy gnraton. h tady flow nrgy quaton, mlarly, can b mlfd for an ntrnally rvrbl (and, hnc, rvrbl) ytm, roducng, a aralll grahcal ntrrtaton for th xtrnal work tranfr. de dt, Stady Stat ( ) = m j j + Q W nt. rv. xt, nt. rv. ( ) ( ) ( ) W xt, nt. rv. = m dj + m d = m d dj = m vd d k d Q nt. m dq nt.rv m Fg. 4.6 For an ntrnally rvrbl ytm, th ara undr th dagram can b aocatd wth th hat tranfr. rv W xt, nt. rv. = m vd + ( k k ) + ( ) (4.8) Whn th chang n k and btwn th nlt and xt ar nglgbl, th quaton mlf to xt, nt. rv. m vd kw [ ] W = (4.9) Notc th mladng rmblanc btwn th quaton and th dv formula, Eq. (.5), for boundary work nvolvng a clod ytm. Whl th xtrnal work for an ntrnally rvrbl ytm roortonal to th ara rojctd on th ax ( Fg. 4.7), th dv work for clod ytm roortonal to th rojctd ara on th v ax 4-

11 on a v dagram. Alo not that an ntrnally rvrbl ytm allow hat tranfr and do not hav to b adabatc. Equaton (4.9) and (4.7) lnd mortant hycal manng to thrmodynamc lot. Evn wthout knowng th dtal of how a dvc actually roduc work or tranfr hat, rmarkabl concluon can b drawn from th grahcal ntrrtaton of th quaton. For ntanc, ga and vaor hav much largr v comard to lqud. Othr aramtr rmanng unchangd, th ntgral n Eq. (4.9) can b xctd to b much largr n magntud for a ga or a vaor comard to a lqud. A comror that ra th rur of vaor, thrfor, wll rqur much mor work than a um rang th rur of a lqud by th am amount (comar Fg. 4.7 and 4.8). In th Rankn cycl ntroducd n Fg..34, th turbn and um orat wth th am rur dffrnc, yt th turbn roduc much mor owr than what th um conum. Onc agan, Eq. (4.9) at th root of th corrct hycal xlanaton Introc Effcncy Sytm can b dalzd n many dffrnt way. Snc frcton, quantfd by ntroy gnraton, downgrad th rformanc of mot dvc, an dal dvc can b aumd to b ntrnally rvrbl,.., S gn,nt =. Obvouly, dvc that rly on frcton cannot b ntrnally rvrbl vn at thr dal lmt. Many ngl-flow dvc uch a um, comror, turbn, nozzl, dffur, tc., can b rgardd a adabatc (no hat tranfr) and orat acro fxd nlt and xt rur undr dgn condton. W wll rfr to th nlt tat a tat- and th xt tat a tat-. At thr dal lmt, th dvc can b condrd ntrnally rvrbl. Rrntng th dal xt tat by tat- 3, th actual and dal dvc can b comard through an ntroy analy a follow. W xt,nt.rv m Fg. 4.7 For an ntrnally rvrbl ytm, th hadd ara roortonal to th magntud of xtrnal work tranfr. ν Actual dvc: ds dt, tady tat ( ) = m + nc S gn,nt Q, adabatc B + S gn,nt (4.) W xt,nt.rv. m Idal dvc: ds dt, tady tat = 3 ( ) = m + 3 Q, adabatc B + S gn,nt, ntrnally rvrbl (4.) h dal ngl-flow dvc, thrfor, mly calld an ntroc dvc. Of cour, t mathmatcally obl for ntroy to rman unchangd btwn th nlt and xt Fg. 4.8 Comard to Fg. 4.7, th work tranfr much l hr bcau th workng flud a lqud wth a much lowr cfc volum. ν 4-

12 f ntroy tranfr through hat lo xactly offt by ntrnal ntroy gnraton; howvr, uch tuaton ar not vry lkly and an ntroc dvc ml that t tady, adabatc, and ntrnally rvrbl. On a lot ( Fg. 4.9), uch a dvc rrntd by th vrtcal old ln jonng th nlt and dal xt tat. For th corrondng actual dvc (rrntd by th dahd ln), ntroy gnraton bng non-ngatv, and th actual xt tat, whch on th am contant rur ln a tat- n Fg. 4.9, hftd to th rght. h ntroc ffcncy comar an nrgtc trm (work outut, work nut, xt kntc nrgy, tc.) from th nrgy quaton of th actual dvc wth th corrondng trm of th ntroc dvc. An Enrgtc rm of Actual (or Idal) Dvc η dvc = (4.) Corrondng rm of Idal (or Actual) Dvc h xact dfnton, whch dvc dndnt, nur that th maxmum obl valu of th ntroc ffcncy do not xcd %. A w dcu ndvdual dvc n th comng cton, w wll ntroduc th dvc-cfc dfnton. 4. Comrhnv Analy In what follow, w wll dcu vral on dvc, ubjctng thm to a comrhnv analy ung th ma, nrgy, and ntroy quaton. Lmtng th dcuon of th nnr workng to a mnmum, w wll ntroduc ach dvc through t functonalt wth mha on ytm-urroundng ntracton. Sutabl aumton and aroxmaton wll b mad to mlfy th govrnng quaton to a t of quaton cutomzd for th artcular dvc. = 3 = 3 Fg. 4.9 h ntroc and actual xt tat har th am xt rur. W xt 4.. P, Duct or ub A, duct, or a tub to rduc confuon, w wll mly u th trm a - a hollow cylndr ud to conduct a lqud, ga, or fnly dvdd old artcl ( Anm. 4.A.). A av a t may aar, a ml can b turnd nto ntrtng dvc uch a a har dryr (Anm. 4.A.harDryr), a watr hatr (Anm. 4.A.watrHatr), a bolr tub, tc., dndng on hat and work ntracton wth th urroundng. Condr a tady flow through a contant damtr of a crtan lngth wth work and hat ntracton a hown n Fg. 4.. If wall frcton n th not Q Fg. 4. Many uful dvc can b rrntd a a flow wth hat and xtrnal work tranfr. 4-

13 nglgbl, how wll t mact th xt vlocty? Bfor w jum to any concluon, lt u whr th balanc quaton lad u. A tady tat ma balanc btwn th nlt and xt yld m = AV / v = A V / v, or V / v = V / v. hrfor, for an ncomrbl flud any flud that can b tratd by th SL modl th ma quaton lad to V = V. hat, th vlocty mut rman contant rgardl of frcton or orntaton. h a owrful concluon whn you thnk about lqud watr flowng downward through a vrtcal. For a (comrbl) ga or vaor flow, howvr, th tuaton bcom comlcatd nc v a functon of tmratur and rur (for an dal ga v = R / ). From th fr body dagram ktchd n Fg. 4., t can b dducd that th rur dcra along th flow to ovrcom frctonal rtanc ( A > A ) rgardl of flud dnty, or hat tranfr. hrfor, n th ca of othrmal flow ( = contant ) or flow wth otv hat tranfr ( ncra), v ncra along th flow nc v = R / for an dal ga (whch alo aroxmatly tru for a vaor). An ncra n cfc volum rult n an ncra n vlocty (rcall from ma quaton that V / v = V / v ). In fact, advancd ga dynamc analy can b ud to how that th ga vlocty can ncra u to th d of ound dt th rnc of frcton n a hatd ga flow. V A Dd you know? h longt undrwatr ln, xtndng mor than 5 ml, carr natural ga from Norwgan North Sa ga fld to Franc. F µ A = A + F µ < Fg. 4. Fr body dagram of th flud n a traght. h forc balanc ndndnt of nrgy or ntroy balanc. V A EXAMPLE 4- [mee] Analy of a Duct Flow. Hlum flow tadly nto a long and narrow adabatc tub of damtr cm wth a vlocty of m/ at kpa and 3 o C. At th xt, th rur dro to 5 kpa to ovrcom wall frcton. Dtrmn th (a) ma flow rat, (b) xt vlocty, (c) xt tmratur, and (d) th rat of ntroy gnraton n th ytm unvr. Nglct chang n kntc and otntal nrg. What-f cnaro: () What would th xt tmratur and vlocty b f kntc nrgy wa not nglctd? SOLUION Analyz th on tady ytm, th unvr nclod wthn th rd boundary of Fg. 4., ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, rfct ga bhavor for hlum, unform tat bad on LE at th nlt and xt, and nglgbl chang n k and. Analy From abl C- or any PG damon, obtan PG modl contant R =.78 / K and c = 5.96 / K for hlum. 4-3

14 h nrgy quaton, Eq. (4.), for th ytm hown n Fg. 4. can b mlfd a follow. de dt ( ) ( ) = m j j + Q W m h h + Q xt ( ) W xt h h = c = ; = = 33 K; Snc h a functon of only for both th PG and IG modl, th concluon that rman contant n an nthalc flow alo vald for any dal ga. h ma quaton, m = m = m, could wth th alcaton of IG quaton of tat, v = R, roduc m AV v AV R 5 ( )( )( ) (.785)( 33) 4 ( 4.99 )(.785)( 33) ( )( 5) = = = = mv mr V = = = = ds dt 5 A A ; m 6.67 h ntroy quaton, Eq. (4.3), for th ytm unvr yld, tady tat Q = m ( ) + S = m ( ) = m c + S gn,unv ln gn,unv R = ln 3. ; 4 kw K Snc th ytm adabatc, thr no xtrnal ntroy gnraton. ; (4.3) t Analy Launch th ngl-flow PG damon. Slct hlum (H) a th workng flud. Calculat th nlt tat, tat-, from th known rort. For th xt tat (tat- ), mak Vl an unknown, ntr, and t mdot = mdot, h = h, and A = A. Calculat to fnd th xt tmratur and vlocty. In th dvc anl, lct th nlt and xt tat, ntr Wdot_xt a zro, and Calculat to fnd Sdot_gn. h manual rult ar rroducd. Fg. 4. Schmatc and = kpa = 5 kpa dagram for Ex

15 What-f Scnaro o nclud th ffct of chang n kntc nrgy, u j = j ntad of h = h n tat-. Sur-Calculat to udat all calculaton. h xt tmratur and vlocty rman vrtually unchangd at 9.97 dg-c and 6.66 m/ rctvly. Dcuon Dt th rnc of frcton, t th vlocty, not tmratur, whch ncra. Although k m to hav no mact n th roblm, j h cannot b automatcally aumd. Intad, u ES to jutfy th nglct of k or on a ca by ca ba. EXAMPLE 4-3 [mee] Analy of a Har Dryr. A har dryr can b modld a a tady flow, whrby a mall fan ull th ar n and forc t through lctrcal rtor whr t hatd a hown n th accomanyng fgur. Ar ntr at th ambnt condton of kpa, 5 o C at a vlocty m/ and lav wth nglgbl chang of rur at a tmratur of 5 o C. h cro-ctonal ara 5 cm. Hat lot from th dryr at a rat of 5 W. Dtrmn th (a) xt vlocty of th flow, (b) rat of lctrcal owr conumton, and (c) th rat of ntroy gnraton n th ytm unvr. U th PG-modl for ar. What-f cnaro: (d) What would th anwr b f th IG modl wa ud? SOLUION Analyz th on tady ytm, th har dryr unvr nclod wthn th rd boundary n Fg. 4.3, ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, rfct ga bhavor of ar, unform tat at th nlt and xt bad on LE, and nglgbl chang n k and. Analy From abl C- or any PG damon, obtan R =.87 / K and c =.5 / K for ar. A n th ca of th rvou xaml, th ma flow quaton, Eq. (4.), could wth th dal ga quaton of tat roduc m A V AV AV v v R ( )( )( ) ( )( ) (.585)(.87)( 33) ( )( ) = = = = = V mv mr ; m.85 ;.5 = = = = A A Wth chang n k and aumd nglgbl, th nrgy quaton, Eq. (4.), yld kpa o 5 C Fg. 4.3 Schmatc and Q = 5W = kpa kpa o 5 C dagram for Ex

16 de dt, tady tat l ( ) ( ) = m j j + Q W m h h + Q W xt ( ) ( )( )( ) ( ) W = mc + Q = =.5 kw; h owr calculatd nclud th owr conumd by th fan. h ntroy quaton, Eq. (4.3), roduc th ntroy gnraton rat n th ytm unvr. ds dt, tady tat Q = m + + S ( ) S gn,unv = m c R gn,unv Q ln ln + =. 49 kw K ; l (4.4) t Analy Launch th ngl-flow PG damon. Slct ar a th workng flud. Calculat tat- and a dcrbd n th ES-cod (otd n ES.Examl ag). h xt vlocty calculatd a art of tat-. In th dvc anl, lct th nlt and xt tat, ntr Qdot, and Calculat to vrfy Wdot_xt and Sdot_gn. What-f Scnaro Launch th ngl-flow IG damon on a arat browr tab. Coy th ES-cod gnratd by th PG oluton nto th I/O anl, and clck th Load button. Rult for th IG modl can b n to b almot dntcal to th PG rult bcau th tmratur varaton of th ga not larg nough to cau any gnfcant chang n th cfc hat. Dcuon An ntroc ffcncy for a har dryr do not mak any n bcau th dal dvc cannot orat ntrocally (rtanc hatng nhrntly gnrat ntroy through lctrcal and thrmal frcton). Howvr, an nrgtc ffcncy ( th nrgy flow dagram of Fg. 4.4) can b dfnd and valuatd a follow. η ( j ) Enrgy Utlzd m j.47 Enrgy Conumd. 5 I = = = = ( W l ) 96. 7% W =.5 kw Ṡ Ṡ gn Q =.5 kw J =.47 kw Ṡ Fg. 4.4 Enrgy and ntroy flow dagram for Ex k ρ + + = contant Fg. 4.5 Brnaull quaton rqur a flow to b tady, ondmnonal, ncomrbl, and ntrnally rvrbl wth no xtrnal work tranfr. Brnoull Equaton For ntrnally rvrbl flow (or rvrbl), w hav alrady n how th xron for xtrnal work and hat tranfr mlfy. It turn out that th nlt 4-6

17 and xt condton can alo b rlatd by a ml formula, known n flud mchanc a th Brnoull quaton. Condr a tady on-dmnonal flow through a varabl damtr ( Fg. 4.5) wth no xtrnal work tranfr. For an ncomrbl, ntrnally rvrbl flow, Eq. (4.8) rduc to xt, nt.rv. = + k k + ρ W = m vd + ( k k ) + ( ) ( ) ( ) V gz V gz + + = + + ρ ( J/) ( J/) ρ ( J/) ( J/ ) (4.5) h th wll known Brnoull quaton rlatng any two flow tat n a ondmnonal ncomrbl flow. Not that th flow do not hav to b adabatc a long a hat tranfr, f any, do not ntroduc any rrvrblt. If th workng flud a ga or a vaor, th otntal nrgy oftn nglgbl and an ncra n vlocty (caud by a rducton n flow ara nc AV = AV ) mut accomany a dcra n rur. h xlan th vntur ffct, whr a contrcton n th flow ara cau a rur dro to ub-atmohrc lvl and atmohrc rur drv th lqud through th traw n Fg. 4.6, caung a ray. Many othr alcaton of Brnoull quaton can b found n any tandard flud mchanc txtbook. kpa 95 kpa kpa Fg. 4.6 h lqud from th traw ray nto th flowng ga du to vntur ffct. > > 4.. Nozzl and Dffur A nozzl a cally dgnd varabl ara duct wth th uro of ncrang th flow vlocty at th xn of a rur dro ( Fg. 4.7 and Anm. 4.A.nozzl). Gnrally a nozzl orat adabatcally btwn known nlt condton, and, and a known xt rur,. Wth no hat or xtrnal work tranfr, th ngl-flow nrgy quaton, Eq. (4.), aum a vry ml form. Fg. 4.7 Nozzl ar convrgng or convrgng/dvrgng varabl ara duct. = mj mj + Q W xt ; j j = (4.6) 4-7

18 hat, th cfc flow nrgy rman contant along th flow n an adabatc nozzl. Nglctng any chang n and ralzng that k much largr than k (rcall th uro of a nozzl), a ml xron for th xt vlocty can b obtand a follow. ( )( ) j = j ; h h + k ; V J/ h h ; (4.7) Clarly, t th nthaly dffrnc that drv th xt kntc nrgy; howvr, h cannot b ndndntly controlld for a gvn nozzl. Mot nozzl ar contourd n a mooth mannr to mnmz frctonal lo o a to maxmz th xt kntc nrgy. An dal nozzl can b xctd to atfy all th condton tady, adabatc, and ntrnally rvrbl to b condrd ntroc. By ttng xtrnal work and chang n otntal nrgy to zro n Eq. (4.8), w can obtan an xron for th xt kntc nrgy of an ntroc nozzl wth th ubcrt rrntng th ntroc xt tat. Introc nozzl: k = h h = vd (4.8) From th quaton, t can b n that d mut b ngatv, that, a rur dro mut occur for th xt vlocty to b gratr than nlt vlocty th hghr th rur dro, th hghr th xt kntc nrgy. W can obtan a ml clod-form xron for ntroc xt vlocty f th workng flud can b modld by th SL or th PG modl. For a lqud nozzl, alcaton of SL modl (contant v ) n Eq. (4.8) rult n. ( )( ) J/ m k = vd = ; V = ; ρ ρ (4.9) h xt vlocty, thrfor, drvn by th quar root of th rur dffrnc btwn th nlt and xt. For a rfct ga, th ntroc rlaton, Eq. (3.64), can b ald, roducng ( k ) / k k = h h = c ( ) = c ; (4.) h xt vlocty n a ga nozzl drvn by th rur rato rathr than th rur dffrnc. h tmratur mut dro along wth rur along th flow followng th 4-8 h Dd you know? h xhaut vlocty of a chmcal rockt about 4 km/. k.. = k = k η nozzl = k Fg. 4.8 Grahcal ntrrtaton of nozzl ffcncy ( Anm. 4.A.nozzlEff.

19 ntroc rlaton (Eq. 3.64). For a gvn rur rato, a nozzl wth a hghr nlt tmratur wll roduc gratr xt vlocty, whch alo vdnt from Eq. (4.8), rdctng a hghr xt kntc nrgy for a workng flud wth hghr avrag cfc volum. Evn f th workng flud an dal ga or a vaor, th concluon aly n a qualtatv n. In a lqud-fuld rockt ngn, hydrogn and oxygn mut b gntd to crat a hgh xt vlocty (and, hnc, thrut). A cold flow, oratng undr th am rur dffrnc, wll roduc a much mallr xt vlocty (and thrut) du to mallr valu of avrag cfc volum. Ung th ntroc nozzl a a bnchmark, th rformanc of an actual nozzl can b maurd by t ntroc ffcncy, dfnd a ( Anm. 4.A.ntrocNozzl) η nozzl.. ( k ) ( k ) KE m j h h h = = KE m j h h h (4.) Sktchd n Fg. 4.8, a h dagram, whch qut mlar to th dagram for th IG or PG modl, mor convnnt for nozzl flow nc th dffrnc n nthal can b drctly ntrrtd a th kntc nrgy. For an actual adabatc nozzl, a tablhd by Eq. (4.), and th xt kntc nrgy can b n to b rducd n Fg W wll rvt nozzl n chatr 5, and dduc th ha of ntroc nozzl through advancd analy. At th ont t uffcnt to mnton that a ubonc nozzl, whch ha an xt vlocty V a, whr a th d of ound at th cro cton wth th mallt ara known a th throat of th nozzl, ha a convrgng contour whl a uronc nozzl wth a V > a ha a convrgng-dvrgng contour a hown n Fg Wll contourd nozzl hav tycal ffcnc of 9% or mor. A dffur work n an xact oot mannr to a nozzl ( Fg. 4.9 and Anm. 4.A.dffurEff); t uro to ncra th rur of a flow at th xn of t kntc nrgy. A a rult, th kntc nrgy at th nlt of a dffur much gratr than th xt kntc nrgy, whch can b nglctd. Alo, th xt rur not fxd and dnd on th rrvrblt rnt n th dffur. A n a nozzl, th rrvrblt ar quantfd through th dffur ffcncy. o dfn th ntroc ffcncy, lt u nglct th xt kntc nrgy (a mor rfnd analy wll b carrd out n chatr 5). Wth no hat or xtrnal work tranfr, th ngl-flow nrgy quaton, Eq. (4.), for th actual and corrondng ntroc dffur can b wrttn a Actual: j = j ; h h + k ; (4.) 4-9 > V < V < a Fg. 4.9 A dffur an xact oot of a nozzl. It ncra rur at th xn of kntc nrgy.

20 Introc: j = j ; h h + k = h ; (4.3) Not that th xt nthal of th actual and ntroc dffur ar qual, whch ndcatd by th horzontal dottd ln n Fg. 4. connctng tat and. Howvr, th ntroc xt tat not ud n dfnng th dffur ffcncy. Intad, a dffrnt xt tat, tat- ' n Fg. 4., whch ha th am rur a th actual xt rur, ' =, and ntroc to tat-, ' =, wth nglgbl kntc nrgy, k ', ud a th targt xt tat for a gvn dffur. Evrythng l at th nlt rmanng unchangd, th rducd nlt kntc nrgy, k ', rqurd to rach th targt xt tat, tat- ', can b obtand from th nrgy quaton. j = j ; h + k = h + k ; k h h ; (4.4) ' ' ', ' ' ' ' h ntroc ffcncy thn dfnd ( Anm. 4.A.dffurEff) a th rato of and k, both of whch roduc th am xt rur η dffur k j h h h = k j h h h ' ' '. (4.5) hu, a 9% ffcncy man that wth 9% of th nlt kntc nrgy, th actual xt rur can b roducd by an ntroc dffur. A th dffur ffcncy dcra, tat- ' n Fg. 4. mov down along th contant ntroy ln and tat- hft to th rght (tat and rman nvarant for a gvn nlt condton); howvr, both tat tll blong to th am contant rur ln at th lowrd xt rur. k ' h k.. = k = = k = η dffur k Fg. 4. Introc ffcncy for a dffur ( Anm. 4.A.dffurEff. EXAMPLE 4-4 [me] Analy of a Nozzl Ar at.5 MPa, 3 o C ntr an nulatd nozzl wth a vlocty of m/ and lav at a rur of. MPa and a vlocty of m/. (a) Dtrmn th xt tmratur of ar. Aum ar to bhav a a rfct ga. (b) I th nozzl ntroc? SOLUION Analyz th on tady ytm, th nozzl nclod wthn th rd boundary n Fg. 4., ung th ma and nrgy balanc quaton. Aumton Stady tat, PG modl for ar wth R =.87 / K and c =.5 / K (abl C.), unform tat at th nlt and xt bad on LE, and nglgbl. Analy 4-

21 Ung th PG modl to valuat nthaly dffrnc, th nrgy quaton for th nozzl, Eq. (4.7), can b mlfd to roduc th xt tmratur. j h + k = h + k ; h h = k k ; k k c ( ) = k k ; = c = = = j ; o 3. C; ( )(.5) h chang n ntroy btwn th nlt and xt can b valuatd from Eq. (3.63). = = c ln R ln = (.5) ln (.87) ln =.4 ; K h only way ntroy can ncra n an adabatc, tady nozzl, through ntroy gnraton; thrfor, th nozzl clarly not ntroc. t Analy Launch th ngl-flow PG damon, and lct ar a th workng flud. Calculat tat- from,, and Vl, and tat- from, Vl, and j = j. and ar valuatd a art of th xt tat. Dcuon Ung th damon, you can dtrmn th ha of th ntroc nozzl rlatv to a gvn nlt ara (aum t to b m ). Snc rur contnuouly dcra along th nozzl, an ntrmdat locaton btwn th nlt and xt can b dntfd by a rur btwn.5 MPa and. MPa. o fnd an ntrmdat tat, ay, tat-3, ck a 3, and ntr 3, mdot3 = mdot, j3 = j, and 3 =. Calculat to fnd A3. Rat wth dffrnt valu of 3 untl you can dduc th ha of th nozzl (convrgng or convrgng/dvrgng). EXAMPLE 4-5 [mee] Analy of a Nozzl Stam at.5 MPa, 5 o C ntr an adabatc nozzl wth a vlocty of 3 m/ and lav at.3 MPa, o C. If th ma flow rat 5 /, dtrmn th (a) nlt and xt ara, (b) xt vlocty, (c) ntroc xt vlocty, and (d) ntroc ffcncy. =.5 MPa Fg. 4. Schmatc and th nozzl n Ex =. MPa dagram for 4-

22 SOLUION Analyz th on tady ytm, th nozzl nclod wthn th rd boundary n Fg. 4., ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, PC modl for tam, unform tat bad on LE at th nlt and xt, nglgbl nlt k, and nglgbl. Analy U ES or th manual aroach to dtrmn th anchor tat tat- for th nlt, tat- for th actual xt, and tat-3 for th ntroc xt ( Fg. 4.). Stat- (gvn,, V and ṁ ) 3 m v =.4744 ; h = 96.6 ; = 7.7 ; K 3 m j = h + k = = 96. ; A = = 79 cm Stat- (gvn,, and ṁ ) 3 m v =.763 ; h = 865 ; ( V / v ) h nrgy quaton for th nozzl, Eq. (4.7), roduc m V = ( k ) = ( j h ) = ( 96. ) = 438 ; hrfor, m A = = 8 cm ( V / v ) For th ntroc xt tat, tat-3, th xt rur and ntroy ar known. Alo th nrgy quaton for th adabatc nozzl roduc j 3 = j. =.5 MPa,3 =.3 MPa 3 Stat-3 (gvn 3 =, 3 =, and j3 = j ) Onc agan, alyng Eq. (4.7) ( ) ( ) ( ) V = k = j h = 4. = m 478 Fg. 4. Schmatc and th nozzl n Ex dagram for h ntroc ffcncy can now b calculatd from Eq. (4.). k k 438 η = = = = 84.% nozzl k k

23 t Analy Launch th ngl-flow PC damon and lct HO a th workng flud. Calculat tat,, and 3 a dcrbd n th ES-cod (otd n ES.Examl). o calculat th ntroc ffcncy, k3 and k mut b manually calculatd n th I/O anl by ntrng xron = Vl3^/ and = Vl^/ rctvly. Dcuon From th t analy, comar th xt ara of th actual and ntroc nozzl, A = 8 cm and A3 = 74 cm. A nozzl contour that do not follow th ntroc ha can cau dd and vortc n an othrw tramlnd flow, rultng n ntroy gnraton (ntrnal rrvrblt) aocatd wth vcou frcton. EXAMPLE 4-6 [mee] Analy of a Dffur Hlum at kpa, 3 K ntr a 9% ffcnt adabatc dffur wth a vlocty of m/. If th xt vlocty nglgbl, dtrmn th xt (a) tmratur, and (c) rur. SOLUION Analyz th on tady ytm, th dffur nclod wthn th rd boundary n Fg. 4.3, ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, PG modl for hlum, unform tat bad on LE at th nlt and xt, and nglgbl. Analy From abl C-, or any PG damon obtan R =.785 / K, c = 5.96 / K, and k =.667 for hlum. h nrgy quaton, Eq.(4.), could wth th PG modl, ald btwn th actual nlt and xt tat, tat- and tat- n Fg. 4.3, yld th xt tmratur a follow. = + h k ; c ( ) j j ; h k + = k V hrfor, = + = 3 + = K; c 5.96 ( )( ) h 3 4. = = Fg. 4.3 Sytm chmatc and th dagram for Ex = kpa h xt rur tll unknown. Howvr, th maxmum obl xt rur can b found by valuatng th ntroc xt tat, tat-3, wth V 3 =. h nrgy quaton wth j3 = j, and V 3 = roduc 3 =. h ntroc rlaton, Eq. (3.64), for th PG modl, roduc

24 ( k )/ k k /( k ) = 3 = = 3. kpa; Ung th dfnton of th dffur ffcncy, Eq. (4.5), w can rlat th actual xt tat, tat-, wth th ntroc xt tat, tat-4, at th actual xt rur ( Fg. 4.3) and obtan = 4. η h h ; = η ; ( ) ( ) 4 dffur 4 3 dffur h3 h ( ) ( ) k / k k / k 4 3 ηdffur = ; Exct for 4, all othr varabl n th quaton ar known. Solvng, w obtan 4 = =.9 kpa. t Analy Launch th ngl-flow PG damon, and lct hlum a th workng flud. Calculat tat,, 3 and 4 a dcrbd on th ES-cod (otd n ES.Examl). Stat-, bng th actual xt tat, contan th anwr. Dcuon h tmratur ncra n a dffur and dcra n a nozzl whn th workng flud a ga or a vaor. For a lqud, howvr, ntroy bng a functon of tmratur only (SL modl), ntroc flow alo ml othrmal flow urbn A turbn a rotary dvc whch dlvr haft work W h at th xn of flow nrgy of th workng flud ( Anm. 4.A.turbn). h nlt condton, and, and th xt rur ar gnrally known for a gvn turbn ( Fg. 4.4). o mlfy analy, chang n k and ar oftn nglctd. Although an actual turbn not rvrbl, Eq. (4.9) can b ud a a gud to xlor t dalzd bhavor. Accordng to th quaton, th work roducd roortonal to th ma flow rat ṁ and th ntgral of vd - th hadd ara n Fg Clarly, th outut can b ncrad f th avrag cfc volum of th workng flud can b ncrad. For a gvn ma flow rat and rur dffrnc, a vaor turbn, thrfor, wll roduc mor work than a hydraulc turbn wth lqud watr a 4-4, k v = c W xt,nt.rv = m Fg. 4.4 Work outut of an ntrnally rvrbl turbn roortonal to th hadd ara. W h vd v

25 th workng flud. For a ga or vaor turbn, an ncra n th nlt tmratur would ncra th avrag cfc volum and, hnc, th outut. hr ar thr man catgor of turbn bad on th workng flud tam (vaor) turbn, ga turbn, and hydraulc (watr) turbn. In a tam turbn, tam xt at a ub-atmohrc rur to a dvc calld condnr (dcud n c. 4..7). In a ga turbn, th xhaut rur atmohrc nc th ga normally xlld to th atmohr. Hat tranfr from turbn undrabl and uually mnmzd ung good nulaton. ycal nlt, xt, and th ntroc xt tat of a tam turbn ar hown n th dagram of Fg h govrnng quaton, Eq. (4.) and (4.3), for an actual and th corrondng ntroc turbn can b mlfd a follow ( ) ( ) Actual: W xt = W h = m j j m h h ; = + S gn / m (4.6) ( ) ( ) Introc: W = m j j m h h ; = (4.7) h, Ung th rult, th ntroc turbn ffcncy, whch comar th rformanc of th actual turbn wth th corrondng ntroc turbn, can b tablhd a ( Anm. 4.A.turbnEff) η W j j h h h turbn = W h, j j h h (4.8) h ntroc tat can b valuatd from a known = and =. Knowldg of th ntroc ffcncy rlat h to a known h, and th actual xt tat can b valuatd from and h. For a ga or vaor turbn, a dro n rur alo accoman a dro n tmratur and ncra n cfc volum (u ntroc PG rlaton a a gud). Wth no gnfcant chang n vlocty, th xt ara mut b gratr than th nlt ara of a vaor or ga turbn. In a tam turbn, th vaor qualty kt uffcntly hgh, 85% or hghr, at th turbn xt to avod turbn damag. h quaton rntd abov ar alcabl to both vaor and ga turbn. For a hydraulc turbn hown n Fg. 4.6 th nlt and xt rur and vloct ar not known. Howvr, th analy can b mlfd by tratng th urfac on th two d of a dam ( Fg. 4.7) a th nlt and xt tat. Equaton (4.6) and (4.8) tll aly 4-5 Fg. 4.5 tam turbn. = = dagram for an actual and ntroc Fg. 4.6 A hydraulc turbn.

26 a long a w do not aum that j h. Whl k can b nglctd du to larg urfac ara, th chang n thn bcom th drvng forc and mut b ncludd n th nrgy quaton. Snc = = ( Fg. 4.6), alcaton of SL modl roduc, ( ) ( ) ( ) ( ) ( ) j j = h h + = u u + + ρ ( ) ( ) / ( J/ ) [ /] j j = c + g z z (4.9) v h outut of an actual hydraulc turbn, W = m ( j j ) h, thrfor, dnd on th avalabl had z z and th tmratur chang brought about by frcton. For an ntroc turbn, = ml = for th SL modl. hrfor, th owr of an ntroc hydraulc turbn rduc to th ml xron W h, = mg ( z z ) /, whch can b drctly dducd from Eq. (4.8). Snc th owr outut dnd manly on th roduct of th ma flow rat and th avalabl had, t obl to roduc condrabl amount of owr vn wth a mall had f th avalabl ma flow rat vry larg. z urbn Fg. 4.7 h drvng forc for a hydraulc turbn th dffrnc n th avalabl had z z. z W h EXAMPLE 4-7 [mee] Analy of a Vaor urbn A tam turbn orat tadly wth a ma flow rat of 5 /. h tam ntr th turbn at 5 kpa, 4 o C and lav at a rur of 7.5 kpa and a qualty of.95. Nglctng th chang n k and a wll a any hat lo from th turbn to th urroundng, dtrmn (a) th owr dvlod by th turbn, (b) th ntroc ffcncy, and (c) th ntroy gnraton rat. What-f cnaro: (d) What would th anwr b f th tam lft th turbn a aturatd vaor? SOLUION Analyz th on tady ytm, th turbn nclod wthn th rd boundary n Fg. 4.8, ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, PC modl for tam, unform tat bad on LE at th nlt and xt, and nglgbl chang n k and. Analy Rrntng th nlt, xt, and th ntroc xt tat by tat, and 3 rctvly, u ES or th manual aroach to obtan th followng tat rort. J,3 W = J J h J 4-6 Fg. 4.8 Sytm chmatc and th nrgy flow dagram for Ex. 4-7.

27 Stat- (gvn, ): j h = 37.8 ; = Stat- (gvn, x ): j h = ; = Stat-3 (gvn 3 =, 3 = ): j3 h3 = 49. ; ; K ; K h nrgy quaton, Eq. (4.6), for th actual and dal turbn yld ( ) ( )( ) Actual: W = m j j = = kw; h ( ) ( )( ) Introc: W m j j kw; h, = 3 = = 43 h ntroc ffcncy now can b valuatd a W η h turbn W = h, = 97. % h ntroy balanc quaton, Eq. (4.3), ald on th turbn unvr roduc = 5 kpa = 7.5 kpa ds dt, tady tat Q = m ( ) + + S gn,unv kw S gn,unv = m ( ) = ( 5)( ) =.37 ; K t Analy Launch th ngl-flow PC damon and lct HO a th workng flud. Evaluat th thr tat a dcrbd n th ES-cod (otd n ES.Examl). Draw th dagram ( Fg. 4.9) ung th lot mnu. In th dvc anl, t u dvc-a a th actual turbn wth tat- and tat- a th anchor tat. Entr Qdot = and Calculat. h haft work and ntroy gnraton rat ar calculatd. Smlarly, t u dvc-b a th ntroc turbn. Obtan th ntroc ffcncy by fndng th rato of Wdot_xt from th two dvc. What-f Scnaro Chang x to n tat-, and Sur-Calculat. Powr outut n th two dvc anl ar udatd to kw and 43.4 kw rctvly, rultng n a rducd ntroc ffcncy of 8.9%. h ntroy gnraton rat ncra gnfcantly to.9 kw/k. Fg. 4.9 dagram for Ex

28 , whch Dcuon An nrgtc ffcncy for th turbn can b dfnd a Wh / ( J J ) mut b % nc th turbn adabatc ( th nrgy dagram ktchd n Fg. 4.8). h ntroc ffcncy, clarly, a bttr maur of th turbn rformanc, tllng u how much otntal outut lot to ntrnal rrvrblt. EXAMPLE 4-8 [mee] Analy of a Hydraulc urbn Watr ntr th ntak of a hydraulc turbn at kpa, 5 o C, m/, and xt at a ont 5 m blow th ntak at kpa, m/. If th turbn outut 7 kw for a flow rat of /, dtrmn (a) th xt tmratur and (b) th ntroc ffcncy. SOLUION Analyz th on tady ytm, th turbn nclod wthn th rd boundary n Fg. 4.3, ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, SL modl for watr, unform tat bad on LE at th nlt and xt. Nglgbl hat tranfr. Analy From abl A- or any SL damon obtan ρ = 997 /m 3 and c v = 4.87 / K for lqud watr. h nrgy quaton, Eq. (4.), could wth th SL modl yld h = ( ) = ( ) + ( + ) m ( ) ( ) ( ) ( k k ) W m j j m h h m k = m u u + + m + m ρ { ( )} = mc v ( ) + m W.986 h o = + = 5. 8 C; mc v cv h ntroc owr can b obtand by ubttutng = n th xron for turbn outut. ( ) m W h, = + m ( ) + m ( k k ) = 98.6 kw ρ 4-8 urbn 5m Fg. 4.3 Schmatc for Ex. 4-8.

29 h ntroc ffcncy, thrfor, can b obtand a W 7 η = h,actual turbn W = h,ntroc 98.6 = 7.3% t Analy Launch th ngl-flow SL damon. Slct watr a th workng flud. Calculat tat, and 3 and analyz dvc A and B a dcrbd on th ES-cod ( ES.Examl). Dcuon Although th tmratur ncra may aar ngnfcant, t account for a gnfcant (about3%) lo of turbn owr du to ntrnal rrvrblt Comror, Fan and Pum Comror, fan, blowr and um ra th rur of a flud at th xn of uful work, whch uually dlvrd through a haft. h dtncton among th dvc tm from th flud thy handl a vaor or a ga for comror, fan, and blowr, and a lqud for um. Whl a fan or blowr ncra th rur of a ga jut nough to crat a drd ma flow, a comror caabl of dlvrng a ga at a vry hgh rur. From our dcuon n c. 4..3, w would xct a comror to rqur mor owr (comar Fg. 4.7 and 4.8) than a um for a gvn ma flow rat and r n rur. A n a turbn, th nlt condton, and, and th xt rur ar gnrally gvn. Alo, chang n k and ar qut mall comard to th chang n flow nrgy j, and ar gnrally nglctd. Dffrnt ty of comror and um ar clafd n Fg. 4.3 and llutratd n Anm. 4.A.comror. h mot common ty th rotary ty, n whch th rotatng blad (calld mllr) mart kntc nrgy to th ncomng workng flud. h hgh d flud a through a dffur cton whr t low down, rultng n an ncra n rur. In an axal flow dvc, th drcton of flow aralll to th rotatng haft whl n a cntrfugal dvc th flud ntr axally through th cntral hub and xt n th radal drcton. In a rcrocatng dvc, th nlt valv on durng th ntak trok at th nd of whch a fxd volum of flud trad. Durng th comron trok both valv ar clod and th xt valv on aftr th flud rach th drd rur or volum. A rcrocatng dvc untady durng a gvn cycl; howvr, whn avragd ovr th tm roluton of ntrt, th cyclc fluctuaton daar and th dvc can b tratd a an on tady ytm (rad th Cntrfugal Fg. 4.3 Scrn hot from Anm. 4.A.comror. Axal Rcrocatng Rotary 4-9

30 dcuon n c.... on how th rcrocatng hat ngn can b tratd a clod tady ytm). Gvn that th work rqurmnt for comron roortonal th ntgral of vd, t drabl to hav mall valu of cfc volum to mnmz th work rqurd for comron. Snc tmratur ncra durng ntroc comron, hat rjcton th mlt way to cool th ga and rduc t cfc volum. Howvr, du to hgh volum flow rat n axal comror, uffcnt tm not avalabl for any gnfcant hat rjcton. On oluton to that to u multtag comron wth ntrcoolng, whr th hot ga at th nd of th frt tag of comron coold, dally, back to th orgnal nlt tmratur wthout any lo of rur. Rcrocatng comror, on th othr hand, can b qud wth fn or watr jackt to nhanc hat rjcton a mor tm avalabl durng th comron roc. By dfault, howvr, w wll aum all comror and um to b adabatc unl othrw mntond. h govrnng balanc quaton for an adabatc comror look almot dntcal to tho drvd for a turbn, xct th work carr a ngatv gn. ( ) ( ) Actual: W xt = W h = m j j m h h ; = + S gn / m ( ) ( ) xt, h, (4.3) Introc: W = W = m j j m h h ; = (4.3) h ntroc comror ffcncy, thrfor, can b dfnd a ( Anm. 4.A.comrorEff) η W j j h h h, comror/um = W h j j h h (4.3) Comar th xron wth th corrondng dfnton for a turbn. h numrator and dnomnator ar wtchd for a work conumng dvc to nur that th ffcncy qual or l than %. h xron for umng owr can b mlfd by ubttutng Eq. (4.9) for th SL modl nto Eq. (4.3). Furthrmor, for an ntroc um = ml = for a lqud. Nglctng th chang n k and acro th um, th umng owr can b xrd a ( Anm. 4.A.umngSytm) um ( ) ( ) ( ) Actual: W = m j j mc + / ρ (4.33) v 4-3 Axal Dynamc Radal /Cntrfugal Comror and Pum Rotary Potv Dlacmnt Rcrocatng Fg. 4.3 Dffrnt catgor of um and comror.

31 ( ) ( ) Introc: W = m j j m / ρ; (4.34) um,, h tmratur r n an actual um, thrfor, a gn of rrvrblt n th um and ra t owr rqurmnt. EXAMPLE 4-9 [me] Analy of a Pum In Ex. 3-5 dtrmn th owr conumton r unt ma of watr by th um by (a) ncludng and (b) nglctng th ffct of kntc nrgy. SOLUION Analyz th on tady ytm, th umng ytm nclod wthn th rd boundary n Fg. 4.3, ung th ma and nrgy balanc quaton. Aumton Stady tat, SL modl for watr, unform tat bad on LE at th nlt and xt. Introc condton ml that th ytm adabatc. Analy Havng valuatd th dffrnc n nthaly and flow nrgy n Ex. 3-5, w mloy th nrgy quaton for th um, Eq. (4.33), to roduc th umng owr r unt ma. W ( ) h, j j j m = = = If th chang n k nglctd, W h, m. ; = j = h k = h 5 =.9 ( 9.8 ) =.95 ; ES Analy Launch th ngl-flow SL damon, and lct watr. Evaluat th nlt and xt tat, tat and ( ES-cod) ung mdot = / a th ba. In th dvc anl, lct th anchor tat, ntr Qdot =, and Calculat. h umng owr for th ntroc um vrfd. o fnd out th ffct of nglctng kntc nrgy, t Vl = Vl and Sur-Calculat. Dcuon Not that th analy comltly ndndnt of what ty of um ud a long a th nlt and xt condton ar th am. W h Fg. 4.3 Sytm chmatc for Ex EXAMPLE 4- [mee] Analy of Comror 4-3

32 Ar comrd from an nlt condton of kpa, 3 K to an xt rur of kpa by an ntrnally rvrbl comror. Dtrmn th comror owr r unt ma flow rat f th dvc (a) ntroc, (b) olytroc wth n =.3, or (c) othrmal. (d) Aumng comron to b olytroc wth an xonnt of.3 and th ntrcoolr to b dal, dtrmn th dal ntrmdat rur f two tag comron whn ntrcoolng ud. SOLUION Analyz th on tady ytm altrnatv comron dvc - ung th ma, nrgy, and ntroy balanc quaton. Aumton Stady tat, PG modl for ar, unform tat bad on LE at th nlt and xt. Analy Obtan R =.87 and k =.4 for ar from abl C- or any PG damon. U th ntroc rlaton, Eq. (3.64), to mlfy th nrgy quaton, Eq. (4.3), for th ntroc comror wth th nlt and xt rrntd by tat- and tat- rctvly. ( ) ( ) ( ) W = m j j m h h = mc h,- ( ) ( ) k / k k / k mkr k = mc = = 8.5 kw; For th olytroc comror, lt tat-3 rrnt th xt tat ( 3 = ). Rcall from c that a olytroc rlaton ha th am form a th corrondng ntroc rlaton k rlacd by n. hrfor, W ( n ) / n mnr 6.6 kw; h,-3 = = n For th othrmal rvrbl comror, oratng btwn tat- and tat-4 = ), th nrgy and ntroy quaton roduc ( 4 ( ) ( ) W = m j j + Q m h h + Q h, ( ) = mc 4 + Q = Q Fg. 4.3 Comror wth ntrcoolr n oton (d) n Ex. 4-. Q 4 can b valuatd from th ntroy quaton a follow. 4-3

33 Q = m + + S 4 ( ) 4 gn 4 hrfor, Q 4 = m c ln W h,-4 = Q 4 = 98.3 kw; ln R = mr ln 4 h am rult, of cour, could b obtand by alyng Eq. (4.9). For th two tag comron ( Fg. 4.3), lt u rrnt th ntrmdat rur by x ( Fg. 4.33). Snc th nlt tmratur th am for ach tag, th xron for th olytroc owr can b ald to ach tag, roducng W W W ( n )/ n ( n ) / n mnr x 7 h, = h,-5 + h,6-7 = + n x o dtrmn x that wll mnmz th total work, w dffrntat th xron wth rct to and t t to zro. A lttl manulaton of th rultng quaton roduc x x = 7 = 36.3 kpa. Subttutng th rult n th abov xron, w obtan W h, = 7. kw. t Analy Launch th ngl-flow PG damon and lct ar a th workng flud. Calculat tat -7 a dcrbd n th ES-cod ( ES.Examl). Notc how th olytroc rlaton ud to ntr v3, v5 and v7. For two tag comron wth ntrcoolng, 5 arbtrarly aumd to b 5 kpa ntally. wo dvc dvc-d and E rrnt th two tag. h um of Wdot_xt for th two tag calculatd a 5 kw. Now chang 5 to a nw valu, Sur-Calculat and fnd th nt owr. Rat untl th otmum rur found. Dcuon h othrmal comror rqur th lat amount of work for a gvn comron rato, whch alo vdnt from th v dagram of Fg (mnmum hadd ara). On way to aroach th othrmal lmt to ndfntly ncra th numbr of tag of ntrcoolng. In ractc, th numbr of tag dcdd by balancng th addtonal cot agant th margnal owr avng rultng from a nw tag Iobarc ntrcoolr Introc Polytroc Introc Polytroc v Iothrmal = MPa = x = kpa Iothrmal Fg v and dagram for dffrnt comror n Ex. 4-. Work conumton th lat for an othrmal comror. 4-33

34 4..5 hrottlng Valv A throttlng dvc a cal rtrcton - an orfc n a lat, a orou lug, a callary tub, or an adjutabl valv ( Fg and Anm. 4.A.xanonValv) dgnd to nhanc frctonal rtanc to a flow n ordr to crat a larg rur dro. = h = contant = Orfc Callary Porou Plug Fg Dffrnt ty of throttlng dvc. h xanon valv llutratd n Anm. 4.A.throttlngValv. Paradoxcally, th flow vlocty omtm ncra (du to an ncra n cfc volum), although th chang n k not gnfcant. hr no xtrnal work tranfr, and hat tranfr alo nglgbl (du to nulaton and hgh ma flow rat). For th adabatc, ngl-flow, on tady dvc, Eq. (4.) and (4.3) mlfy a Enrgy: j = j ; h h ; (4.35) IG/PG modl = h = contant Entroy: = + S gn / m ; > ; (4.36) hrottlng, a can b n from th nrgy quaton, can b rgardd a an nthalc roc. If th workng flud an dal (or rfct) ga, nthaly bng a functon of tmratur only, w cannot xct any tmratur chang whn a ga throttld. Howvr, for a PC flud, th concluon can b dratcally dffrnt. A can b n from th dagram ( Fg. 4.35) of a PC flud, th contant-nthaly ln nd th aturaton dom hav ngatv lo. o vrfy th, draw a contant nthaly ln n th lot anl aftr valuatng any aturatd lqud tat ung a PC tat damon. h man that f a aturatd lqud throttld, th nthalc rqurmnt roduc a aturatd mxtur at th xt that mut b at a much lowr tmratur a hown n Fg = Fg Contant nthaly ln on th dagram rval f thr a tmratur chang n throttlng.