Introduction to Thermodynamics

Transcription

1 Unestà d sa Intoduton to hemodynams.. Intoduton. Hstoy of of hemodynams.. he he Fst Fst Law. Mosop ew. Joule he he eond Law. Mosop ew. Canot hemodynam opetes of of Fluds Multomponent ystems

2 Unestà d sa Intoduton D deals wth equlbum states. Knowng ntal ondtons t pedts how the system wll end up, but t annot say how long t wll take. D was deeloped wth steam engnes,.e. mahnes that onet heat nto wok (Watt, 760 s). Natue of heat: a) alo, a onseed quantty: hot objets ontan moe alo (Laose, 770 s, Laplae); b) Intenal moements (B. hompson, 80 s). he poduton of mote powe n steam engnes s due to the tanspotaton of alo fom a wam body to a old body (ad Canot, 84). Canot postulated that some alo s lost, not beng oneted nto mehanal wok. hs s the bass of the seond law, whh theefoe pedates the undestandng of the fst law. Conseaton of mehanal enegy (.e. knet plus potental) was fomulated by Newton. Joule (940 s) demonstated the equalene between heat and wok ( al. 4.9J). Clausus (Rudolf Gottleb) ntodued the onept of ntenal enegy (850) and of entopy (86). Boltzmann (890 s) showed how themodynams an be deed fom statstal mehans (loss of nfomaton auses entopy nease).

3 Unestà d sa he fst law 4 3 u ~ N u E u ~ ; u~ 0 N N mu ; E M M u~ ; u~ N u ~ Mehanal Enegy, E K Coheent moton Note : Joule: wok and heat ae two foms of enegy exhange. aloe Joule (J N m Kg m s - ) N mu ; M No Intenal Momentum Intenal Enegy, U Inoheent moton Wok podues oheent patle moement. Heat podues noheent patle moement. δq du δw δq U law: otal enegy s onseed δw Q 0; WMg z U m ; 0Kg 0 ms - 0 m / Kg J Kg - K - 0.5K (dd Joule heat?) 3

4 Unestà d sa he eond Law () ) ~ 4 3 ~ 4 3 E K U U U 3 tme u ~ N tme a) u ~ N hese eents ae so mpobable that they ae patally mpossble (puttng thngs n ode s moe dffult than the opposte) ) 4 3 tme 3) 4 law: Enegy tends to dspese. a) Dstbutes unfomly n spae. u ~ N b) b) Moes fom ts oheent to ts noheent foms. Keenan s fomulaton of the law: an solated system tends to eah ts state of stable equlbum, oespondng to maxmum enegy dspeson. Equpatton theoem: at equlbum eah degee of feedom has the same mean enegy k/, whee k s Boltzmann onstant and defnes tempeatue. In deal gas, eah patle has enegy 3k/. 4

5 Unestà d sa he eond Law () Clausus: No poess s possble whose sole esult s the tansfe of heat fom a oole to a hotte body (poess a). H Q H Q C W W Q H - Q C w Q C / W Q C / (Q H - Q C ) Keln: No poess s possble whose sole esult s the oneson of heat fom a eseo nto wok (poess b). C H Q H W w < C / ( H C ) η W / Q H - Q C / Q H Q C η < - C / H H H C Q C Q H W Q H W Equalene of the two fomulatons. Q C Q C C C Note: the les aboe ndate that we ae efeng to poesses, n whh a wokng flud undegoes a themodynam yle at the end of whh t s bought bak to ts ntal ondtons. In geneal, the seond law states that t s mpossble that the sole esult of a tansfomaton s to ompletely onet heat nto wok, o tansfe heat fom old to hot. 5

6 Unestà d sa oblems. A powe plant buns hydoabons and podues 000 MWe, wth a 40% effeny. a) How muh does t onsume? How muh of that powe s dshaged nto the old eseo (.e. the sea)? b) How many ltes of hydoabons does t buns pe hou? (Assume that the fuel podues 4 MJ/kg, wth a densty of 0.9 g/m 3 ) ) If the same powe wee to be podued n a hydoelet powe plant, opeatng though a 000 m heght dffeene, alulate the equed olumet flux. Answe. a) Q H W / η 03 MW / MW. Q C Q H W 500 MW. b) Fuel equed J/s 3600 s/h / (4 0 6 J/kg 0.9 kg/lt) lt/h gal/h. (.5 mllon ltes oespond to a ontane 0m 0m.5m) ) A e hang a m 3 /s flow ate would podue a powe W mg z m 3 /s 0 3 kg/m m/s 00 m MW. heefoe, podung 000 MW eques a 000 m 3 /s. (moe o less, that of the Msssspp e). Consde an ntenal ombuston engne. Hee the gas extats heat fom a sngle eseo (.e. the flame), and expands, mong a pston and podung wok. Does t ontadt the seond law? Answe. No. he seond law states that t s mpossble that the sole esult of a tansfomaton s to ompletely onet heat nto wok, o tansfe heat fom old to hot. Hee at the end of the tansfomaton the gas s not n the same ondtons as t wee at the begnnng and theefoe the seond law s not applable. 6

7 Unestà d sa he eond Law (3) B C Q ABC H Q C ADC δq A C C A d C A d δq entopy A D δw Fdx d; d Adx F / A; pessue F A du δq δw d d dx U ntenal enegy H U ; dh d d H enthalpy A U ; da d d A Helmholtz fee enegy G H ; dg d d G Gbbs fee enegy U onst. when and ae onst. H onst. when and ae onst. A onst. when and ae onst. G onst. when and ae onst. hemal equlbum: Mehanal equlbum: onstant onstant 7

8 Unestà d sa hemodynam opetes of Fluds () N n n/6 s the numbe of ollson pe unt sufae and unt tme. m s the momentum tansfeed to the wall n eah ollson. ( ) u nu~ 6 Ideal Gas N mu~ m u ~ 3 N k N N N A N # moles N A Aogado # NR o R an de Waals: Z R 3 3 3k/m (equpatton theoem) o: Real Gas / Lqud 9 8 Z R b R a whee R N A k s the gas onstant Compessblty fato C C C (edued tempeatue) (edued pessue) (edued olume) Law of oespondng states: Z Z(, ) s unesal, ald fo any mateal. Wth small oetons, t woks emakably well. ee plot next page. 8

9 Unestà d sa hemodynam opetes of Fluds () he Compessblty Fato Z /R Redued essue 9

10 0 Unestà d sa hemodynam opetes of Fluds (3). - ompessblty sothemal. olumeexpansty κ β ( ) d d d d d β κ dg da dh U U U U U d d du ; ; ; : the fnd,,, Usng. ; equatons Maxwell Usng the Maxwell equatons, the aaton of any themodynam quantty fo a sngle omponent system an be expessed n tems of the aaton of and (o ), knowng only β, κ and (o ) U heat apaty at onstant olume H heat apaty at onstant pessue ( ) ( ) ; depends on path) ( heat apaty dh Q du Q Q Q δ δ δ δ

11 Unestà d sa oblem d d d d d d d β ), ( a) a) Fnd d(,); b) Fnd d(,); ) Fnd. d d d d d d d κ β ), ( b) ( ) ( ) κ β : In fat ( ) κ β κ β κ β β d d d d d d d ; ) Note : Aodng to the equpatton pnple (.e. the seond law), fo a monoatom gas U 3 / k N A, so that 3 / R (R N A k). In geneal, lassal physs pedts that U / N, whee N s the numbe of degees of feedom of mole of omponent, so that s a onstant. Explanng why t s not so eques quantum physs. Note : Fo an deal gas, R/, so that β / and κ /. Consequently, R. Altenately: fo an deal gas, U U(); H U U R H(), so that du/d and dh/d, wth R.

12 Unestà d sa Multomponent ystems () du δq δw N d d N ( ) µ dn G N µ dn µ hemal potental of th omponent N numbe of moles of th omponent dg, µ µ s the mola fee enegy of th omponent wthn mxtue G N g G N g hemal equlbum: onstant Mehanal equlbum: onstant Chemal equlbum: µ onstant GN µ N µ haseα:,, x, x,, x N- haseβ:,, x, x,, x N- Eah phase s desbed by, (same n eah phase) plus ompostons x N /N (N- ndependent x ) Gbbs phase ule. (N # omponents; π # phases; F degees of feedom) aables:, and x n eah phase. # aables: (N-). Independent elatons: µ α µ β µ π fo eah omponent. # elatons: N (π-). F -π N

13 Unestà d sa ngle Component, wo hase ystems α ap Lage knet enegy, small potental enegy N ; π ; F so: () sat (). β lq mall knet enegy, lage potental enegy At equlbum,, ae the same. o, dung phase tanston (soba and sothemal), G onst., so that g α g β. (gg/n). On aeage, moleules n the two phases hae the same enegy. β lq Ctal pont dg α dg β (, ) (, ) (, ) (, ) s α d α d s β d β d α ap sat d s s h h d β α β α ( ) β α β α ln sat β (apo), α l (lqud) l << R/ (deal gas) d d sat h R l / h l Latent heat of apozaton h l sat d ln R d(/ ) (Clausus-Clapeyon) ald at low 3

14 Unestà d sa Multomponent ystems () Q Ideal gas s Q W f d R f d R ln f ( ) d Rd( ln ) dg g R ln f (fugaty) deal gas : f Lqud : f f sat ( ) f ( ) ap,, sat, f sat Lq,, sat, f sat, x,g, x, g Mxng of deal mxtues x x ; x / s R ( x x x ln x ); g ( h s) s (,, onst.) ln, g g xg R ln ( x ln x x x) xµ xµ g x µ g R ln x R ln ( x f ) 4

15 Unestà d sa Multomponent ystems (3) ap,, y Lq,, x p y x sat ( ) apo Lqud Equlbum µ ap µ lq R ln y f ap Raoul law: ald fo deal mxtues. lq n patula, ald fo deal lqud mxtues,.e. benzene / toluene, omposed of spees hang smla moleules In geneal, Raoult law s ald when x. R ln x f Heny s law: the solublty of a gas n a lqud at a patula tempeatue s popotonal to the pessue of that gas aboe the lqud: p H x H Heny's law onstant In patula, H sat fo deal mxtues. In geneal, Heny s law s ald when x 0. 5