MTX221. Session 40 ENTROPY (CONTROL VOLUME) Sessie 40 ENTROPIE (KONTROLE VOLUME) Dr. Jaco Dirker. These slides also appear on Click-UP

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1 s.40-1 MTX1 ss 40 ENTROPIE (KONTROLE VOLUME) sson 40 ENTROPY (CONTROL VOLUME) Dr. Jaco Drkr Ths slds also appar on Clck-UP Hrd skyfs vrskyn ook op Clck-UP 8 th dton / 8 utgaw Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

2 s.40- Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss ngl flow procsss ar qut common (sngl nlt and sngl outlt) Exampls: Turbn, pump, comprssor tc. Turbn, pomp, komprssors For thos dvcs (whch nvolvs work), th work can b obtand usng th 1 st law and th nd law combnd. (th way w dd t thus far) Thr s howvr anothr approach, usng a custom- constructd quaton: W hav th frst law and scond law xprssons for stady stat sngl nlt- sngl outlt condtons: E q s h s gn 1 1 V gz h V q s T gz w Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

3 s.40-3 Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss W hav: s s gn q T s E q h 1 1 V gz h V gz w Entropy quaton wrttn n dffrntal form: s gn q / T ds Rwrttn as: q Tds T s gn Employng on of th Gbbs quatons: ( Tds dh vdp ): q dh vdp T s gn Aftr ntgraton and substtutng nto th nrgy quaton: w vdp V gz Z 1 V NOT ON FORMULAE HEET Ts gn E Hrd s n altrnatw mtod Ths s an altrnatv mthod of dtrmnng th work trm. (shaft work) Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

4 s.40-4 Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss w vdp V gz Z 1 V What can w notc about ths? Entrop-vrwkkng vrmndr war arbd 1. Th ntropy gnraton trm rducs th amount of ral work that can b xtractd. (Th maxmum work can b xtractd f a rvrsbl procss s followd and ntropy gnraton s zro) Omkrbar pross. Consdr a rvrsbl procss gnorng PE and KE: Whn th prssur ncrass, th work s ngatv (work don on th systm), such as a pump or comprssor. Ts gn E Indn druk tonm s d arbd ngatf Whn th prssur dcrass, th work s postv (work don by th systm), such as a turbn. Indn druk afnm s d arbd postf Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

5 s.40-5 Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss Omkrbaar, gn KE n PE vrandrng For dvcs whch ar opratd n a rvrsbl procss wth lttl chang n potntal and kntc nrgy, th quaton rducs to: 1 w vdp gz Z Ts gn V V E E w vdp NOT ON FORMULAE HEET *** On a Pv dagram ths s gvn by th ara to th lft of th graph. Ths s only vald for spcal cass such as sngl flow dvcs. psal gvall Exampl: Ths s a comprssor or pump: hould not b confusd wth th ntgral usd n pston dvcs. YOU HOULD HAVE EEN THI BEFORE! - EE THE ELFTUDY EXAMPLE 4.6 *** Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

6 s.40-6 Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss ngl Flow Exampl 1: Consdr EX7.7 (Ed 8) Don wth ths nw approach as wll as wth our prvous approachs. Calculat th work pr klogram to pump watr sntropcally from 100 kpa, 30ºC to 5 MPa. Bpaal d arbd pr klogram om watr sntrops t pomp vanaf 100 kpa, 30ºC tot 5 MPa.. wart agtrgrond Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

7 s.40-7 Gstadgd Enklvlo Prosss 7.3 tady tat ngl Flow Procss w vdp V gz Z 1 V Ts gn E A spcal form of th abov drvd quaton s th Brnoull quaton usd wdly n Flud dynamcs. : tromngslr vp 1 1 V gz vp V gz Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

8 s.40-8 Bgnsl van Entrop Tonam 7.4 Prncpl of th Incras of Entropy W alrady consdrd th prncpl of ntropy ncras for a control mass Bhr massa Lt s do ths now for a control volum: Bhr volum Consdr Control Volum A stuatd nsd Control Volum B (Formng an Isolatd systm) D volum rul massa, htt n ntrop ut Volum A and B xchang, mass, nrgy and ntropy m E Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

9 s.40-9 Bgnsl van Entrop Tonam 7.4 Prncpl of th Incras of Entropy Th Entropy Balanc for ths volums ar: Volum A: Volum B: Total: d d Q CV, A m s m s gn, A TA dt Q CV, B m s m s gn, A TA dt d dt nt d dt CV, A d CV, B dt gn, A gn, B 0 Rsultng n: d nt dt 0 THU: Entropy s not consrvd, and has to ncras Entropbhoud bstaan n, n dt mot tonm. Thus, wth tm, t s also found that wth control volums, th nt ntropy ncrass. Entrop tonam vnd ook n n Bhrvolum plaas Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

10 s Bgnsl van Entrop Tonam 7.4 Prncpl of th Incras of Entropy Th ntropy ncrass du to th ntrnally as wll as xtrnally rrvrsblts. onomkrbaarhd On of th most mportant xtrnal rrvrsblts ar du to hat transfr ovr a tmpratur dffrnc. Ekstrn onomkrbaarhd wns htt-oordrag oor n tmp. vrskl Ths gnraton trm s gvn by: Rmmbr: gn, B Q T A Q T B Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

11 s Engnrng Applcatons: Effcncy W know what th dfnton of ffcncy of a cycl: War krnglop Ingnurstopassngs: Rndmnt th Rndmnt dfns W Q nt H (for a hat ngn) Ral cycls hav lowr ffcncs than dal or rvrsbl cycls Lar rndmnt krngloop Can w dfn an ffcncy for a machn? (IE an lmnt that mght b prsnt n a cycl) Exampls: turbns, pumps and comprssors Gas Turbn Radal Comprssor B A Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

12 s.40-1 Ingnurstopassngs: Rndmnt 7.5 Engnrng Applcatons: Effcncy FLAH BACK Rfr to th xampl w dd durng Chaptr 6 about th stam turbn whch lost som hat compard to on that was Adabatc. (Not n txtbook, ssson 4/5 approx.) Turbn ndd to produc 5 MW of Elctrcty: Th adabatc turbn only ndd 5.7 kg/s of stam Th turbn whch lost 1 MW of Hat ndd 6.9 kg/s (whch was lss ffcnt) Th adabatc turbn thus was mor ffctv D adabats turbn was dus mr ffktf Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

13 s Ingnurstopassngs: Rndmnt 7.5 Engnrng Applcatons: Effcncy Idal Turbns oprat adabatcally and th xpanson procss s thus sntropc Ral Turbns, howvr, do loos som hat and hav som frcton, and hav som uncontrolld xpanson whch rsults n lowr work output (Hat transfr s rrvrsbl) Ontwrps-vrandrlks In gnral th followng ar dsgn varabls for turbns: P, T, P (P and T ar prpard n a pror procss, whl P s lmtd by th nvronmnt nto whch th turbn dschargs.) Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

14 s Ingnurstopassngs: Rndmnt 7.5 Engnrng Applcatons: Effcncy Consdr th dffrnc n th procsss of an dal turbn to that of a possbl ral turbn. (Entropy not constant), n ths xampl thr s an ncras n th ntropy: Th xt pont s ndcatd as such: Isntropc procss: s Ral procss: Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

15 s Ingnurstopassngs: Rndmnt 7.5 Engnrng Applcatons: Effcncy Lt us consdr ths furthr: Htt-vrls vroorsaak ntrop afnam If th turbn s not nsulatd, t wll loos hat and thus also loos ntropy as wll (at som rat). Onomkrbaarhd skp ntrop Howvr, th othr rrvrsblts wthn th turbn such as frcton and a masur of uncontrolld xpanson wll rsult n ntropy gnraton (at som rat) In th fgurs blow adabatc condtons ar consdrd, but wth rrvrsblts Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws

16 s Ingnurstopassngs: Rndmnt 7.5 Engnrng Applcatons: Effcncy Rndmnt van n turbn Th ffcncy of a turbn s dfnd va comparson btwn th actual work producd, w (or w a ) and th dal work that mght b producd, (rvrsbl or sntropc) w s : w In a cas whr th turbn s adabatc, and ΔKE and ΔPE ar gnord: turbn w s Gnral 1st Law: q h 1 1 V gz h V gz w E Rducd 1 st Law: (In many cass ΔKE and ΔPE may b gnord.) In gnral turbn ffcncs rang from 0.7 to 0.88 Largr turbns usually ar dsgnd mor ffcntly Dpartmnt of Mchancal and Aronautcal Engnrng Dpartmnt Mgans n Lugvaartkundg Ingnursws turbn w w s h h h h h s h w Th turbn ffcncy can thn also b wrttn as: E w h h Ths s calld th adabatc ffcncy. Adabats rndmnt