The First Law of Thermodynamics: Control Volumes

Transcription

1 Th Frst Law of Throdynacs: Control Volus Hr w wll xtnd th consrvaton of nrgy to systs that nvolv ass flow across thr boundars, control volus. Any arbtrary rgon n spac can b slctd as control volu. Thr ar no concrt ruls for th slcton of control volus. Th boundary of control volu s calld a control surfac. Consrvaton of Mass Lk nrgy, ass s a consrvd proprty, and t cannot b cratd or dstroyd. Mass and nrgy can b convrtd to ach othr accordng to Enstn s forula: E = c, whr c s th spd of lght. Howvr, xcpt for nuclar ractons, th consrvaton of ass prncpl holds for all procsss. For a control volu undrgong a procss, th consrvaton of ass can b statd as: total ass ntrng total ass lavng = nt chang n ass wthn Control volu o Fg. : Consrvaton of ass prncpl for a. Th consrvaton of ass can also b xprssd n th rat for: d / dt Th aount of ass flowng through a cross scton pr unt t s calld th ass flow rat and s dnotd by. Th ass flow rat through a dffrntal ara da s: d = ρv n da whr V n s th vlocty coponnt noral to da. Thus, th ass flow rat for th ntr cross scton s obtand by: M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus

2 A V n da (kg/s) Assung on dnsonal flow, a unfor (avragd or bulk) vlocty can b dfnd: = ρ V A (kg/s) whr V (/s) s th flud vlocty noral to th cross sctonal ara. Th volu of th flud flowng through a cross scton pr unt t s calld th volutrc flow, V : V VndA VA A ( /s) Th ass and volu flow rat ar rlatd by: =ρv = V / v. Consrvaton of Enrgy For control volus, an addtonal chans can chang th nrgy of a syst: ass flow n and out of th control volu. Thrfor, th consrvaton of nrgy for a control volu undrgong a procss can b xprssd as total nrgy crossng boundary as hat and work + total nrgy of ass ntrng total nrgy of ass lavng = nt chang n nrgy of W En, ass Eout, ass E Ths quaton s applcabl to any control volu undrgong any procss. Ths quaton can also b xprssd n rat for: W de, / dt de, / dt de n ass out ass / dt Mass n Control volu W Mass out Fg. : Enrgy contnt of can b changd by ass flow n/out and hat and work ntractons. Work flow: s th nrgy that s rqurd to push flud nto or out of a control volu. Consdr an agnary pston (that pushs th flud to ) whr th flud prssur s P and th cross sctonal ara s A. Th forc actng on th pston s F = PA. M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus

3 A P P Flow drcton Fg. : schatc for flow work. Th work don n pushng th flud s: W flow = F.s = PA.s = PV (kj) or n a unt bass, w flow = W flow / = Pv (kj/kg) Not that th flow work s xprssd n trs of proprts. Th flow work can also b wrttn as a rat quaton. Th flud ntrng or lavng a control volu posssss an addtonal for of nrgy (flow nrgy Pv). Thrfor, th total nrgy of a flowng flud on a unt ass bass (dnotd by θ) bcos: θ = Pv + = Pv + (u + k + p) (kj/kg) Rcall that nthalpy s dfnd as: h = u + Pv. Thrfor, th abov quaton bcos: θ = h + k + p = h + V / + gz (kj/kg) Th proprty θ s calld thalpy. By usng nthalpy nstad of ntrnal nrgy, th nrgy assocatd wth flow work nto/out of control volu s autoatcally takn car of. Ths s th an rason that nthalpy s dfnd! Stady Stat Flow Procss s A procss durng whch a flud flows through a control volu stadly s calld stadystat procss. A larg nubr of dvcs such as turbns, coprssors, and nozzls oprats undr th sa condtons for a long t and can b odld (classfd) as stady flow dvcs. Th tr stady pls no chang wth t. Th tr unfor pls no chang wth locaton ovr a spcfd rgon. A stady flow s charactrzd by th followng: M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus

4 No proprts wthn th chang wth t. Thus, volu, ass, and nrgy of rans constant. As a rsult, th boundary work s zro. Also, total ass ntrng th ust b qual to total ass lavng. No proprts chang at th boundary of th wth t. It ans that th ass flow rat and th proprts of th flud at an opnng ust ran constant durng a stady flow. Th hat and ass ntractons btwn th and ts surroundngs do not chang wth t. Usng th abov obsrvaton, th consrvaton of nrgy prncpl for a gnral stadyflow syst wth ultpl nlts and xts can b wrttn as: W W h V gz h V gz Nozzls and Dffusrs A nozzl s a dvc that ncrass th vlocty of a flud at th xpns of prssur. A dffusr s a dvc that ncrass th prssur of a flud by slowng t down. Th cross sctonal ara of a nozzl dcrass n th flow drcton for subsonc flows and ncras for suprsonc flows. W = 0 Control Volu Nozzl = 0 A > A V > V P < P Fg. : Schatc of nozzl. For a nozzl, coon assuptons and dalzatons ar: = 0, no t for hat transfr, du to hgh vlocty W = 0, nozzls nclud no shaft or lctrc rsstanc wrs ΔPE = 0, no chang n flud lvaton. Mass quaton for nozzls bcos: = or ρ A V = ρ A V M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus

5 Enrgy balanc: W = θ θ (h + V / ) at nlt = (h + V / ) at xt Dffusrs ar xactly th sa dvc as nozzls; th only dffrnc s th drcton of th flow. Thus, all quatons drvd for nozzls hold for dffusrs. W = 0 Control Volu Dffusr A < A V > V P < P = 0 Fg. 5: Schatc for dffusr. Exapl : Nozzl Sta ntrs a convrgng dvrgng nozzl opratng at stady stat wth P = 0.05 MPa, T = 00 C and a vlocty of 0 /s. Th sta flows through th nozzl wth nglgbl hat transfr and no sgnfcant chang n potntal nrgy. At th xt, P = 0.0 MPa, and th vlocty s 665 /s. Th ass flow rat s kg/s. Dtrn th xt ara of th nozzl, n. Assuptons:. Stady stat opraton of th nozzl. Work and hat transfr ar nglgbl, = W = 0.. Chang n potntal nrgy fro nlt to xt s nglgbl, ΔPE = 0. Th xt ara can b calculatd fro th ass flow rat : A = v / V W nd th spcfc volu at stat. So, stat ust b found. Th prssur at th xt s gvn; to fx th stat anothr proprty s rqurd. Fro th nrgy quaton nthalpy can b found: V V h h h h V Fro Tabl A 6, h = 78.9 kj/kg and vlocts ar gvn. V M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 5

6 h N kj 78.9 kj / kg s kg. / s 0 N kj / kg Fro Tabl A 6, wth P = 0.0 MPa, and h = kj/kg, (usng ntrpolaton) on fnds T = 00 C (approxatly) and v = 6.5 /kg. Th xt ara s: kg / s6.5 / kg A / s Turbns and Coprssors Fg. 6: Convrgng dvrgng nozzl. In sta, gas, or hydrolctrc powr plants, th dvc that drvs th lctrc gnrator s turbn. Th work of turbn s postv snc t s don by th flud. Coprssors, pups, and fans ar dvcs usd to ncras th prssur of th flud. Work s suppld to ths dvcs, thus th work tr s ngatv. Coon assuptons for turbns and coprssors: = 0. ΔPE = ΔKE = 0. Exapl : Turbn Sta ntrs a turbn at stady stat wth a ass flow rat of 600 kg/h. Th turbn dvlops a powr output of 000 kw. At th nlt th prssur s 0.05 MPa, th tpratur s 00 C, and th vlocty s 0 /s. At th xt, th prssur s 0 kpa, th qualty s 0.9, and th vlocty s 50 /s. Calculat th rat of hat transfr btwn th turbn and surroundngs, n kw. M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 6

7 Assuptons: Stady stat opraton. Th chang n potntal nrgy s nglgbl. Th nrgy balanc for th turbn s: W W h V gz V h h h V V gz Control Volu Turbn Shaft W = 0 (nglgbl) Fg. 7: Schatc for turbn. Usng Tabl A 6, h = 78.9 kj/kg, usng Tabl A 5 wth x = 0.9, h = h f + x h fg = 9.8 kj/kg + 0.9(9.8 kj/kg) = 5.5 kj/kg Thrfor, h h = = 9.55 kj/kg Th spcfc kntc nrgy chang s: N kj V V kj / kg s kg. / s 0 N. = (000 kw) kg/h ( kj/kg) (h / 600 s) (kw / kj/s) = 6. kw M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 7

8 Not that chang n spcfc kntc nrgy s vry sall. Also s sall copard to W. Th ngatv sgn n ans that hat transfr s fro to surroundngs. Throttlng Valvs Any knd of flow rstrctng dvcs that causs a sgnfcant prssur drop n th flud s calld throttlng valv. Fg. 8: Thr knds of throttlng valvs. Th prssur drop n flud s oftn accopand wth a tpratur drop n fluds. Th agntud of ths tpratur s govrnd by a proprty calld th Joul Thoson coffcnt. Coon assuptons for throttlng valvs: = 0 W = 0 ΔPE = ΔKE = 0. Th consrvaton of nrgy bcos: h = h u + P v = u + P v Th flow nrgy ncrass durng th procss (P v > P v ), and t s don at th xpns of th ntrnal nrgy. Thus, ntrnal nrgy dcrass, whch s usually accopand by a drop n tpratur. M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 8

9 For dal gass h = h(t) and snc h rans constant th tpratur has to ran constant too. Mxtur Chabrs (Drct Contact Hat Exchangrs) Mxtur chabrs ar usd to x two stras of fluds. Basd on ass prncpl, th su of ncong flow quals to su of lavng flud. Control Volu Hot Watr Mxng chabr Cold Watr = 0 W = 0 Fg. 9: T lbow s a xng chabr. Coon assuptons for xng chabr: = 0, W = 0, ΔPE = ΔKE = 0. Th consrvaton of nrgy bcos: Σ( h) nlt = Σ( h) outlt Hat Exchangrs Hat xchangrs ar dvcs whr two ovng flud stras xchang hat wthout xng. Coon assuptons for hat xchangrs: W = 0 ΔPE = ΔKE = 0. could b dffrnt dpndng on th slctd, s th nxt xapl. Th ass and nrgy balanc bco: and h h h Mxd watr h M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 9

10 Flud A Hat Flud B Flud B Flud A Fg. 0: Schatc for hat xchangr. Exapl : Hat xchangr Engn ol s to b coold by watr n a condnsr. Th ngn ol ntrs th condnsr wth a ass flow rat of 6 kg/n at MPa and 70 C and lavs at 5 C. Th coolng watr ntrs at 00 kpa and 5 C and lavs at 5 C. Nglctng any prssur drops; dtrn a) th ass flow rat of th coolng watr rqurd, and b) th hat transfr rat fro th ngn ol to watr. Watr 5 C Control Volu Hat 70 C 5 C Watr 5 C W choos th ntr hat xchangr as our control volu, thus work transfr and hat transfr to th surroundngs wll b zro. Fro ass balanc: W and M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus 0

11 Th consrvaton of nrgy quaton s: W W h h h h W h h V h h h W h gz h V h gz Assung constant spcfc hat for both th ol and watr at thr avrag tpratur, W C C P, P, Watr T T T T kj kg. C kj kg. C 6 kg / n 0. kg / n b) To dtrn th hat transfr fro th ol to watr, choos th followng. Th nrgy quaton bcos: W h C T T P, kj kg. C 6 kg / n kj / kg 5 C Control volu 70 C M. Bahra ENSC 88 (F09) st Law of Throdynacs: Control Volus