Chapter 4. the conservation of. total. energy E. Energy of the System. = Leaving the System (4.1) referred to. Normally. Total Energy. AAiT.

Size: px

Start display at page:

Download "Chapter 4. the conservation of. total. energy E. Energy of the System. = Leaving the System (4.1) referred to. Normally. Total Energy. AAiT."

Martha Stanley
5 years ago
Views:

1 Throdynacs I _ Chatr 4 Th Frst Law of Throdynacs It s th law that rlats th arous fors of nrgs for syst of dffrnt tys. It s sly th xrsson of th consraton of nrgy rcl Th frst law of throdynacs, also known as th consraton of nrgy rcl, rods a sound bass for studyg th rlatonshss aong th arous fors of nrgy and nrgy tractons. Basd on xrntal obsratons, th frst law of throdynacs stats thatt Enrgy can b nthr cratd nor dstroydd durg a rocss; t can only chang fors. A ajor consqunc of th frst law s th xstnc nrgy E. and th dfton of th rorty total Fgur 4.1 Dffrnt fors of nrgs For th syst shown abo, th consraton of throdynacs s xrssd as nrgy rcl or th frst law of Total Enrgy Total Enrgy Th Chang Total = Entrg th Syst Lag th Syst Enrgy of th Syst E E E syst (4.1) Ths rlaton s oftn rfrrd to as th nrgy balanc undrgog any kd of rocss. and s alcabl to any kd of syst Norally th stord nrgy, or total nrgy, of a syst s xrssd as th su of thr sarat nrgs. Th total nrgy of th syst, E sys st, s gn as Cold by Ydnkachw M. Pag 1 of 1

2 Throdynacs I E syst Whr: = Intrnal Enrgy+ Ktc Enrgy+ Potntal Enrgy E syst =U+KE+PE (4.) U s th su of th nrgy contad wth th olculs of th syst and s calld th trnal nrgy. Th ktc nrgy KE and th otntal nrgy PE ar gn by KE = PE gz Or U ( u u ) 1 1 KE ( 1 ) PE g( z z ) 1 Now th consraton of nrgy rcl, or th frst law of throdynacs for closd systs, s wrttn as E E U+ KE+ PE (4.3) If th syst dos not o wth a locty and has no chang laton, t s calld a statonary syst, and th consraton of nrgy quaton rducs to E E U (4.4) Th chanss of nrgy transfr at a syst boundary ar: Hat, Work, ass flow. Only hat and work nrgy transfrs occur at th boundary of a closd (fxd ass) syst. On systs or control olus ha nrgy transfr across th control surfacs by ass flow as wll as hat and work. Cold by Ydnkachw M. Pag of 1

3 Throdynacs I _ 4.1 Mchanss of Enrgy Transfr, E and E ou Hat Transfr, (Q) Hat transfr to a syst (hat ga) crass th nrgy of th olculs and thus th trnal nrgy of th syst and hat transfr fro a syst (hat loss) dcrass t sc th nrgy transfrrd as hat cos fro th nrgy of th olculs of th syst. Q s zro for adabatc systs. Work Transfr, (W) Work transfr to a syst (.., work don on a syst) crass th nrgy of th syst, and work transfr fro a syst (.., work don by th syst) dcrass t, sc th nrgy transfrrd as work cos fro th nrgy contad th syst. Car ngs and hydraulc, sta, or gas turbs roduc work whl corssors, us, and xrs consu work. Mass Flow, () Whn ass ntrs a syst, th nrgy of th syst crass bcaus ass carrs nrgy wth t ( fact, ass s nrgy). Lkws, whn so ass las th syst, th nrgy contad wth th syst dcrass bcaus th lag ass taks so nrgy wth t. Th nrgy balanc can b wrttn or xlctly as ut E E ( Q Q ) ( W W ) ( Eass, Eass, ) E Syst (4.5) E Fgur 4. Th nrgy contn of a control olu can b changd by ass flow as wll as hat and work tractons. E Nt nrgy transfr by hat, work and ass E Syst Chang trnal, ktc, otntal, tc..nrgs Cold by Ydnkachw M. Pag 3 of 1, kj

4 Throdynacs I _ Or on a rat for, as E E Rat of nt nrgy transfr by hat, work and ass E Syst Rat chang trnal, ktc, otntal, tc..nrgs kw For constant rats, th total quantts durg th t tral Δt ar rlatd to th quantts r unt t as Q Q t, W W t and E Et (kjj ) (4.6) Th nrgy balanc ay b xrssd on a r unt asss bass as syst ( kj / kg) (4.7) 4. Th frst law and a closd syst For th closd syst whr th ass nr crosss balanc s Q-Q+W -W = E syst th syst boundary, (4.8) thn th nrgy Closd syst undrgog a cycl For a closd syst undrgog a cycl, th tal and fal stats ar dntcal, and thus Fgur 4.3 For a cycl E = 0, thus Q = W. E syst E E1 0 E E 0 Cold by Ydnkachw M. Pag 4 of 1

5 Throdynacs I E E (4.9) Notg that a closd syst dos not ol any ass flow across ts boundars, th nrgy balanc for a cycl can b xrssd trs of hat and work tractons as W Q or W nt, Q nt, (4.10) nt, nt, If th total nrgy s a cobaton of trnal nrgy, ktc nrgy and otntal nrgy. E U KE PE (4.11) ( 1 ) Q1 W1 U U1 g( Z Z1) (4.1) For nglgbl changs ktc and otntal nrgy Intrnal nrgy and Enthaly Intrnal nrgy Q U U W (4.13) Th trnal nrgy cluds so colx fors of nrgy show u du to translaton, rotaton and braton of olculs. It s dsgnatd by U and t s xtns rorty. Or r unt ass as, scfc trnal nrgy,, s an tns rorty of th syst lk P,, and T but not asurabl. Th trnal nrgy xrsson can b usd to dtr th stat of th substanc f on addtonal rorty s known. If w tak two has as lqud and aor at a gn saturaton rssur or tratur Or U U U (4.14) f g u f u f gug (4.15) Fally u u xu (4.16) f fg Cold by Ydnkachw M. Pag 5 of 1

6 Throdynacs I Enthaly It s anothr xtns rorty whch has a unt of nrgy and t s dnotd by H. Th nthaly s a connnt groug of th trnal nrgy, rssur, and olu and s gn by H U P (4.17) Th nthaly r unt ass s,, whch s sad to b scfc nthaly and h=u+p Consdr a ston cyldr assbly whr w ha a contuous suly hat so that th boundary changs for th rocss s Q U U W (4.18) (In th abo quaton w ar nglctg chang ktc nrgy and otntal nrgy) If w ar assug th rocss s at constant rssur W Pd P( ) 1 1 (4.19) W1 P P1 (4.0) Hnc Q U U P P (4.1) Q U P U P (4.) If w tak two has at a gn saturaton stat Q1 H H1 (4.3) H H H (4.4) f fg h h xh f fg (4.5) 4.3 Scfc Hat It s an tns rorty of a substanc that wll nabl us to coar th nrgy storag caablty of arous substancs. Th unt s. Cold by Ydnkachw M. Pag 6 of 1

7 Throdynacs I It dfd as; th nrgy rqurd to ras th tratur of a unt ass of a substanc by on dgr. In gnral, ths nrgy dnds on how th rocss s xcutd. (hat s ath dndnt rorty) In throdynacs, w ar trstd two kds of scfc hats: scfc hat at constant olu and scfc hat at constant rssur. Th scfc hat at constant olu can b wd as th nrgy rqurd to ras th tratur of th unt ass of a substanc by on dgr as th olu s atad constant. Hr th boundary work s zro bcaus th olu s constant Fro frst law δq du (4.6) Pr unt ass q du but q CdT CdT du (4.7) du C (4.8) dt (Chang trnal nrgy wth tratur at constant olu) Th scfc hat at constant rssur C can b wd as th nrgy rqurd to ras th tratur of th unt ass of a substanc by on dgr as th rssur s atad constant. Fro frst law δq du Pd du P dh (4.9) Pr unt ass q dh, but q CdT CdT dh (4.30) C dh dt (4.31) (Chang nthaly wth tratur at constant rssur) Scfc hats ar sots gn on a olar bass. Thy ar thn dnotd by and and ha th unt kj/kol C or kj/kol K. Cold by Ydnkachw M. Pag 7 of 1

8 Throdynacs I 4.4 Intrnal Enrgy, Enthaly, and Scfc Hats of Idal Gass W dfd an dal gas as a gas whos tratur, rssur, and scfc olu ar rlatd by P RT (4.3) It has bn donstratd athatcally (th cog chatrs) and xrntally that for an dal gas th trnal nrgy s a functon of th tratur only. That s, U U( T) (4.33) Usg th dfton of nthaly, w ha Cobg th abo to quaton hu P but P RT hu RT (4.34) Ths shows that h h( T) Fro th scfc hat rlaton du C ( T ) dt (4.35) u u1 CdT (4.36) Or takg arag alu of scfc hat for narrow tratur dffrnc u u C ( T T ) (4.37) 1 a, 1 By th sa argunt, And, dh C T dt (4.38) h h1 CdT (4.39) h h1 C ( ) a, T T1 (4.40) Hnc, u C T (4.41) hc T (4.4) Cold by Ydnkachw M. Pag 8 of 1

9 Throdynacs I Rlaton btwn C P and C for Idal Gass Usg th dfton of nthaly (h = u + P) and wrtg th dffrntal of nthaly, th rlatonsh btwn th scfc hats for dal gass s hu RT (4.43) dh du RdT (4.44) Rlacg dh by CdTand du bycdt w ha CdTCdT RdT (4.45) C C R (4.46) Or r ol (olar bass) C C R (4.47) Whr s th unrsal gas constant =8.314 KJ/Kol At ths ot, w troduc anothr dal-gas rorty calld th scfc hat rato k, dfd as C K (4.48) C C KC (4.49) Cobg quaton (4.46) and (4.49) KC C R (4.50) C R K 1 (4.51) and C C R (4.5) K C K R K 1 (4.53) Cold by Ydnkachw M. Pag 9 of 1

10 Throdynacs I _ Intrnal Enrgy, Enthaly, and Scfc Hats of Solds and Lquds A substanc whos scfc olu (or dnsty) s constant s calld an corssbl substanc. Th scfc olus of soldss and lquds ssntally ra constant durg a rocss. Thrfor, lquds and solds can b aroxatd as corssbl substancs. It can b athatcally shown that th constant-olu and constant-rssur scfc hats ar dntcal for corssbl substancs Th scfc hat can b xrssd as C C C (4.54) 4.5 Th Frst Law and th Control olu Th consraton of ass and th consraton of nrgy rcls for on systs or control olus aly to systs hag ass crossg th syst boundary or control surfac. In addton to th hat transfr and work crossg th syst boundars, ass carrs nrgy wth t as t crosss th syst boundars. Thus, th ass and nrgy contnt of th on syst ay chang whn ass ntrs or las th control olu. Fgur 4..4 Tycal control olu or on syst Hnc th consraton of ass rcl can b usd to rlat ass whch ntrg and lag a syst. It can b xrssd as Th nt ass transfr to or fro a control olu durg a rocsss (a t tral t) s qual to th nt chang (cras or dcras) th total ass wth th control olu durg that rocss ( t). That s, Cold by Ydnkachw M. Pag 10 of 1

11 Throdynacs I Total ass ntrg Total ass lag Nt chang ass - = th C durg Δt th C durg Δt wth th C durg Δt C ( kg ) (4.55) It can also b xrssd rat for as d / dt C ( kg / s ) (4.56) Whr and ar th total rats of ass flow to and of th control olu, and d C /dt s th t rat of chang of ass wth th control olu boundars. So t w also us olu flow rat whch dcats th olu of th flud flowg through a art r unt t and dnotd by, (4.57) Throdynac rocsss olg control olus can b consdrd two grous: stadyflow rocsss and unstady-flow rocsss. Stady stat rocss Th flow through a control olu s at stady stat f, th rorty of th substanc at a gn oston wth or at th boundars of th control olu do not chang wth t. Durg a stady-flow rocss, th total aount of ass contad wth a control olu dos not chang wth t ( C = constant). Thn th consraton of ass rcl rqurs that th total aount of ass ntrg a control olu qual th total aount of ass lag t. Mass th control olu s constant d C dt C A A 0 corssbl assuton (4.58) (4.59) (4.60) (4.61) (4.6) (4.63) Cold by Ydnkachw M. Pag 11 of 1

12 Throdynacs I _ Unstady stat rocss Th rorts wth th control olu chang wth t but ra unfor at any stant of t. Th rorts at th flow aras do not chang wth t although th ass flow rats ay chang wth t. Tycal xal:- fllg and tg rocsss whr ost of rorts ust b usd. th cass arag alu of For such cass d c dt d c dt And for sgl stras, d c dt 0 (4.64) (4.65) (4.66) 4.6 Flow Work and Th Enrgy of a Flowg Flud Unlk closd systs, control olus ol ass flow acrosss thr boundars, and so work s rqurd to ush th ass to or of th control olu. Ths work s known as th flow work, or flow nrgy, and s ncssary for atag a contuous flow through a control olu. Fgur 4.5 Schatc for flow work Cold by Ydnkachw M. Pag 1 of 1

13 Throdynacs I If th flud rssur s P and th cross-sctonal ara of th flud lnt s A, th forc ald on th flud lnt by th agary ston s F PA (4.67) To ush th ntr flud lnt to th control olu, ths forc ust act through a dstanc L. Thus, th work don ushg th flud lnt across th boundary (.., th flow work) s Wflow FL PAL P Pr unt ass ( kj ) (4.68) wflow P (4.69) Hnc, w P and w P (4.70) flow, flow, xt Rat xrsson of flow work W ( P ) and W ( P ) flow, flow, xt (4.71) Th total flow work s, W P P flow (4.7) Th total work of th syst s W W floww c (4.73) Dlont of nrgy balanc Th gnral rrsntaton of th frst law of throdynacs Q1 W1 E E1 (4.74) Th frst law for on syst wll also ha th sa for, but W1 Wflow Wc (4.75) Cold by Ydnkachw M. Pag 13 of 1

14 Throdynacs I Th total nrgy of a sl corssbl syst conssts of thr arts: trnal, ktc, and otntal nrgs E = Intrnal Enrgy+ Ktc Enrgy+ Potntal Enrgy (4.76) Pr unt ass E =U+KE+PE uk u gz (4.77) Th flud ntrg or lag a control olu osssss an addtonal for of nrgy, th flow nrgy P, as alrady dscussd. Thn th total nrgy of a flowg flud on a unt-ass bass (dnotd by ) bcos P P ( u k ) (4.78) But th cobaton P + u has bn rously dfd as th nthaly h. So th rlaton th abo quaton rducs to hk h gz ( kj / kg ) (4.79) For lt P u gz P (4.80) P h gz (4.81) For lt P h gz (4.8) Gnral quaton E E de / dt syst 0( stady) 0 (4.83) Q W Q W Q W h ( gz) Q W h ( gz) (4.84) (4.85) Cold by Ydnkachw M. Pag 14 of 1

15 Throdynacs I _ In such cass, t s coon ractc to assu hat to b transfrrd to th syst (hat ut) at a rat of, and work roducd by th syst (work ut) at a rat of, and thn sol th robl. Th frst-law or nrgy balanc rlaton that cas for a gnral stady-flow syst bcos QW h h 1 1 g( z 1 ( z ) Ddg by gs th nrgy balanc on a unt-ass bass as ( 4.86) qwh qwh h 1 h So Stady-Flow Engrg Dcs 1 gz ( z ) 1 ( 4.87) Whn th flud xrncs nglgbl changs ts ktc and otntal nrgs (that s, k = 0, = 0), th nrgy balanc quaton s rducd furthr to ( 4.88) Nozzls and Dffusrs Nozzls and dffusrs ar coonly utlzd jt ngs, rockts, saccraft, and n gardn hoss. A nozzl s a dc thatt crass th locty of a flud at th xns of rssur. A dffusr s a dc that crass th rssur of a flud by slowg t down. That s, nozzls and dffusrs rfor oost tasks. Th cross-sctonal ara of a nozzl dcrass th flow drcton for subsonc flows and crass for sursonc flows. Th rrs s tru for dffusrs. Fgur 4.6 Schatc dagras of Nozzls and dffusrs Cold by Ydnkachw M. Pag 15 of 1

94) Turbs In sta, gas, or hydrolctrc owr lants, th dc that drs th lctrcc gnrator s th turb.

16 Throdynacs I _ For flow through nozzls, th hat transfr, work, and otntal nrgy ar norally nglctd, and nozzls ha on ntranc and on xt. Th consraton of nrgy bcos (4.89) 1 (4.90) E E (4.91) Q nt lt h gz W nt xt h gz (4.9) h h (4.93) ( h h ) (4.94) Turbs In sta, gas, or hydrolctrc owr lants, th dc that drs th lctrcc gnrator s th turb. As th flud asss through th turb, work s don agast th blads, whchh ar attachd to th shaft. As a rsult, th shaft rotats, and th turb roducs work. Fgur 4.7 Schatc dagra of Turbs Cold by Ydnkachw M. Pag 16 of 1

17 Throdynacs I _ 1 E E Q nt lt h gz W nt xt h gz h h W (4.95) W h ( h) (4.96) If w nglct th changs ktc and otntal nrgs as flud flows through an adabatc turb hag on ntranc and on xt, th consraton of ass and th stady-stat, stady- flow frst law bcos Corssors Corssors, as wll as fans, ar dcs usd to cras th rssur of a flud. Work s suld to ths dcs fro an xtrnal sourc through a rotatg shaft. Thrfor, corssors ol work uts. En though ths thr dcs functon slarly, thy do dffr th tasks thy rfor. A fan crass th rssur of a gas slghtly and s aly usd to oblz a gas. Fgur 4.8 Schatc dagra of Corssors Cold by Ydnkachw M. Pag 17 of 1

18 Throdynacs I If w nglct th changs ktc and otntal nrgs as flud flows through an adabatc corssor hag on ntranc and on xt, th stady-stat, stady-flow frst law or th consraton of nrgy quaton bcos Qnt h gzw nt h gz lt xt W nt ( h h ) W nt ( h h) (4.97) Pus Th work rqurd whn ug an corssbl lqud an adabatc stady-stat, stadyflow rocss s gn by 1 QW h h1 g( z z1) Th nthaly dffrnc can b wrttn as h h1 u u1 P P 1 (4.98) For corssbl lquds w assu that th dnsty and scfc olu ar constant. Th ug rocss for an corssbl lqud s ssntally sothral, and th trnal nrgy chang s aroxatly zro (w wll s ths or clarly aftr troducg th scond law). Thus, th nthaly dffrnc rducs to th dffrnc th rssur scfc olu roducts. Sc = 1 = th work ut to th u bcos 1 W P P1 g( z z1) (4.99) s th nt work don by th control olu, and t s notd that work s ut to th u; so, =, u If w nglct th changs ktc and otntal nrgs, th u work bcos W P P1 W P P, u 1 (4.100) (4.101) W us ths rsult to calculat th work suld to bolr fd watr us sta owr lants. Cold by Ydnkachw M. Pag 18 of 1

Throdynacs I _ Throttlg als Throttlg als ar any kd of flow-rstrctg dcs that caus a sgnfcant rssur dro th

Unlk turbs, thy roduc a rssur dro wth olg any work.

rfrgraton and ar-condtong alcatons. Fgur 4.9 Schatc dagras of Throttlg als Q nt lt h gz W nt xt h gz h h (4.

19 Throdynacs I _ Throttlg als Throttlg als ar any kd of flow-rstrctg dcs that caus a sgnfcant rssur dro th flud. So falar xals ar ordary adjustabl als, callary tubs, and orous lugs. Unlk turbs, thy roduc a rssur dro wth olg any work. Th rssur dro th flud s oftn accoand by a larg dro tratur, and for that rason throttlg dcs ar coonly usd rfrgraton and ar-condtong alcatons. Fgur 4.9 Schatc dagras of Throttlg als Q nt lt h gz W nt xt h gz h h (4.10) h h (4.103) Mxg Chabrs Th xg of two fluds occurs frquntly ngrg alcatons. Th scton whr th xg rocss taks lac s calld a xg chabr. Th ordary showr s an xal of a xg chabr. Cold by Ydnkachw M. Pag 19 of 1

Nglctg ktc and otntal nrgs and notg th rocss s adabatc wth no work, w ha for two ntrancs and on xt h 1

20 Throdynacs I _ Fgur 4.10 Schatcc dagra of Mxg Chabr 1 3 (4.104) (4.105) 3 1 (4.106) Q nt lt E h E gz W nt xt h gz Accordg to th sktchd control olu, ass crosss th control surfac. Nglctg ktc and otntal nrgs and notg th rocss s adabatc wth no work, w ha for two ntrancs and on xt h 1 1 h 3h 3 h 1 1 ( h h) 3( h h h 3h3 3 ( h h ) 3 ( h h ) 1 3 ) (4.107) (4.108) (4.109) (4.110) Cold by Ydnkachw M. Pag 0 of 1

Th us, fans, and blowrs causg th fluds to flow across th control surfac ar norally locatd sd th control surfac. Fgur 4.

21 Throdynacs I _ Hat Exchangrs Hat xchangrs ar norally wll-sulatd dcs that allow nrgy xchang btwn hot and cold fluds wth xg th fluds. Th us, fans, and blowrs causg th fluds to flow across th control surfac ar norally locatd sd th control surfac. Fgur 4.11 A hat xchangr can b as sl as two concntrc s. (4.111) For ach flud stra sc thr s no xg. 1 3 w 4 R (4.11) (4.113) Q nt lt h E gz E W nt xt h gz h 1 1 3h3 w ( h 1 h) R ( h h3) h 4h4 4 (4.114) (4.115) Cold by Ydnkachw M. Pag 1 of 1

CHAPTER 4. The First Law of Thermodynamics for Control Volumes

CHAPTER 4. The First Law of Thermodynamics for Control Volumes CHAPTER 4 T Frst Law of Trodynacs for Control olus CONSERATION OF MASS Consrvaton of ass: Mass, lk nrgy, s a consrvd proprty, and t cannot b cratd or dstroyd durng a procss. Closd systs: T ass of t syst