Second Law. first draft 9/23/04, second Sept Oct 2005 minor changes 2006, used spell check, expanded example

Transcription

1 Second Law firs draf 9/3/4, second Sep Oc 5 minor changes 6, used spell check, expanded example Kelvin-Planck: I is impossible o consruc a device ha will operae in a cycle and produce no effec oher han he raising of a weigh and he exchange of hea wih a single reservoir. Clausius: I is impossible o consruc a device ha operaes in a cycle and produces no oher effec han he ransfer of hea from a cooler body o a hoer body. Woud: used o: ) predic he direcion of processes ) esablish he condiions of final equilibrium 3) deermine bes possible heoreical performance of a process if i is impossible o have a hea engine wih % efficiency, how high can i go?? define ideal process, ermed reversible process: a process ha, once having aken place, can be reversed wihou changing eiher he sysem or surroundings examples irreversible; pison expanding agains sop reversible; pison expanding by removing and replacing weighs; excerp from VW&S page 66 good descripion of reversible and irreversible processes Le us illusrae he significance of his definiion for a gas conained in a cylinder ha is fied wih a pison. Consider firs Fig. 6.8, in which a gas (which we define as he sysem) a high pressure is resrained by a pison ha is secured by a pin. When he pin is removed, he pison is raised and forced abruply agains he sops. Some work is done by he sysem, since he pison has been raised a cerain amoun. Suppose we wish o resore he sysem o is iniial sae. One way of doing his would be o exer a force on he pison, hus compressing he gas unil he pin could again be insered in he pison. Since he pressure on he face of he pison is greaer on he reurn sroke han on he iniial sroke, he work done on he gas in his reverse process is greaer han he work done by he gas in he iniial process. An amoun of hea mus be ransferred from he gas during he reverse sroke in order ha he sysem have he same inernal energy i had originally. hus he sysem is resored o is iniial sae, bu he surroundings have changed by virue of he fac ha work was required o force he pison down and hea was ransferred o he surroundings. hus he iniial process is an irreversible one because i could no be reversed wihou leaving a change in he surroundings. In Fig. 6.9 le he gas in he cylinder comprise he sysem and le he pison be loaded wih a number of weighs. Le he weighs be slid off horizonally one a a ime, allowing he gas o expand and do work in raising he weighs ha remain on he pison. As. he size of he weighs is made smaller and heir number is increased, we approach a process ha can be reversed, for a each level of he pison during he reverse process here will be a small weigh ha is exacly a he level of he plaform and hus can be placed on he plaform wihou requiring work. In he limi, herefore, as he weighs become very small, he reverse process can be accomplished in such a manner ha boh he sysem and surroundings are in exacly he same sae hey were iniially. Such a process is a reversible process. 9/5/6

2 Carno cycle example seam power plan - working subsance seam boiler - hea ransferred from high (consan) reservoir o seam - seam only infiniesimally lower han reservoir => reversible isohermal hea ransfer process. (phase change fluid - vapor is such a process urbine - reversible adiabaic (no hea ransfer) decreases from H o L condenser - hea rejeced from working fluid o L reservoir (infiniesimal Δ) some seam condensed pump - emperaure raised o H adiabaicly can reverse and ac as refrigeraor Carno cycle four basic processes:. reversible isohermal process in which hea is ransferred o or from he H reservoir. reversible adiabaic process in which he emperaure of he working fluid decreases from H o L 3. reversible isohermal process in which hea is ransferred o or from he L reservoir 4. reversible adiabaic process in which he emperaure of he working fluid increases from L o H Carno cycle mos efficien, and only funcion of emperaure efficiency (in hea engine) W = energy_sough Q H Q L Q L η hermal = QH = energy_ha_coss = Q H = QH emperaure scale (arbirary bu defined in erms of Carno efficiency) Q H f ( H ) H = = proposed by L Q H Q L f ( η L Lord Kelvin hermal = L ) H η hermal = Q L = ψ( L, H ) mos efficien a his poin have raio of absolue emperaures derive scale from non-carno hea engine operaing a seam H and ice emperaure L if we could measure i would find η o be 6.8% H η h =.68 = L if wan difference o be as on he Celsius scale Δ := H := L := Given L iniial values.68 = H = L + Δ H 9/5/6

3 H L := Find ( H, L ) _deg_c = _deg_k H L = VW&S has 73.5 changed o 73.6 o correspond o riple poin of waer. deg_c Enropy inequaliy of Clausius... for fig 7. dq dq = Q H Q L > from definiion of absolue emperaure scale and H and L consan Q H Q L Q = d = H L if.. dq approaches, H approaches L, while reversible dq = => for all reversible hea engines... dq and... dq = if irreversible, wih H, L, and Q H same... W irrev < W rev Q H Q L = W for boh Q H Q L_irrev < Q H Q L_rev => Q L_irrev > Q L_rev dq = Q H Q L_irrev > and... if hea engine becomes more irreversible such ha W =>.. Q H Q L_irrev dq = < H L as... dq = dq < => all irreversible engines dq dq < should do refrigeraion cycle as well 9/5/6 3

4 example figure 7.3 pg 88 VW&S example fig simple seam power plan cycle - no ypical - pump handles mixure of liquid and vapor in such proporions ha sauraed liquid leaves he pump and eners he boiler. he pressures and qualiy a various poins are given in he figure.? Does his daa saisfy he inequaliy of Clausius? Sauraed vapor,.7 MPa Boiler urbine W inequaliy of Clausius... dq hea is ransferred in boiler and condenser, boh a consan 9% qualiy, 5 kpa - sauraed liquid,.7 MPa Condenser 4 Pump % qualiy, 5kPa 3 dq = dq boiler + dq condenser = dq boiler + dq condenser boiler condenser on a per uni mass basis mass := kg MPa := 6 Pa kj := 3 J kpa := 3 Pa kj boiler... p :=.7MPa h fg := 66.3 kg := deg_c seam ables able A.. kj q _ := h fg q _ = 66.3 kg q = Δh from firs law kj kj p condenser = p 3 = p 4 = 5kPa h f := 5.94 h kg fg := 373. kg 3 := seam ables able A.. x 3 :=.9 h 3 := h f + x 3 h fg kj x 4 :=. h 4 := h f + x 4 h fg q 3_4 := h 4 h 3 q 3_4 = kg _deg_c = _deg_k q _ q 3_4 kj in_dq_over_ := + in_dq_over_ =.87 deg_k is < kg example figure 7.3 pg 88 VW&S 9/5/6 4

5 enropy plo daa.5 wo reversible cycles from o (no labeled) A - B and... A - C dq = reversible... A process B process C process A -B A - C dq = = dq + A + dq B dq = = dq A dq C A B A C subrac second from firs => ds = dq B = dq C B C δq rev reversible... dqrev. = S S so as we did for energy E (e) in firs law dq is independen of pah in reversible process => is a propery of he subsance enropy is an exensive propery and enropy per uni mass is = S (7.3) (7.), W (.3) 9/5/6 5

6 enropy change in a reversible process example Carno Carno cycle four basic processes:. reversible isohermal process in which hea is ransferred o or from he H reservoir. reversible adiabaic process in which he emperaure of he working fluid decreases from H o L 3. reversible isohermal process in which hea is ransferred o or from he L reservoir 4. reversible adiabaic process in which he emperaure of he working fluid increases from L o H plo daa o i :=.. Q _ dq rev. = S S = H o 3 - adiabaic S i := o 4 3 dqrev. = = S 3 S S 3 = S 4 i := Q 3_4 dq rev. = S S = L 3 4 o - adiabaic S dqrev. = = S S 4 S = S 4 i := oal cycle... i :=.. 5 S S S in general.. for reversible process, area under S curve represens Q 9/5/6 6

7 wo relaionships for simple compressible subsance: Gibbs equaions in Woud from firs law... δq = de + δw wihou KE or PE δq = du + δw (5.4) reversible... δq = d δw = p dv as.. e.g. a pison... δw = F d = pa ds = pdv subsiue... ds = du + p d QED (7.5) since... H = U + p dh = du + p dv + V dp ds = du + p δv on a per uni mass... subsiue... ds = dh V dp QED (7.6) ds = dh v dp applicable o BOH reversible and irreversible processes as hey are relaionships beween sae variables enropy change for irreversible process plo daa (7.7).5 reversible cycle from o o (no labeled) A - B and... irreversible cycle from o o (no labeled) A - C.5 A - B reversible... d Q = = d Q A + A A - C irreversible subrac second from firs and rearrange d Q B B A process reversible B process reversible C process irreversible dq = dq A + d Q C < inequaliy of Clausius A C dq C dq A + dq B dq A + > A B A C 9/5/6 7

8 dq B dq C > dq B > dq C B C B C dq B_rev = ds B_rev = ds C B_rev is a propery and alhough calculaed for reversible process, is idenical beween saes for irreversible. subsiue ino inequaliy above... equaliy holds when ds C > dq δq reversible and when C or in general... ds S S dq C irreversible, he change of enropy will be greaer han he reversible principle of increase in enropy (W.4) consider sysem a and surroundings a o, δq ransferred from surroundings o sysem due o above δq ds sysem for he surroundings, δq is negaive herefore ds surr = δq oal ne change in enropy is... δq δq ds ne = ds sysem + ds surr = δq since hea is ransferred FROM surroundings, o > herefore... hus... principle of increase in enropy ds δq ne if > o reverse signs and resul holds ds ne = ds sysem + ds surr for all processes ha a sysem and is surroundings can undergo ds isolaed_sysem 9/5/6 8

9 second law for a conrol volume no developed bu second law saed in erms of los work LW δq δlw ds = + during δ change in enropy is... S S δq δlw = + δ δ δ (7.43) S = enropy_in_c_v_a_ime_ S _δ = enropy_in_c_v_a_ime plus_δ S = S + s δm i i = enropy_of_sysem_a_ime_ S = S _δ + s δm e e = enropy_of_sysem_a_ime plus_δ S S = S _δ S + s δm e e s δm i i (7.44) ec... second law for a conrol volume d Sc_v + (m_do e s e ) (m_do i s i ) Q_do c_v d n n c_v (7.49) = when reversible seady sae, seady flow process d Sc_v = (7.5) d Q_do c_v (m_do e s e ) (m_do i s i ) (7.5) = when reversible n n c_v 9/5/6 9

10 uniform sae, uniform flow process rewrie 7.49 as... d d (ms) c_v ( ) ( ) + m_do e s e m_do i s i n n c_v and inegrae... d (m s) d d c_v = m s m s Q_do c_v m = ( m_do i s i s i )d = ( i ) ( m_do e s e ) d ( m e s e ) n n n n (7.54) = when reversible in conrol volume Q_do c_v herefore for ime m s m s + ( m e s e ) (m i s i ) d (7.55) = when reversible n n c_v since he emperaure over he conrol volume is uniform a any insan of ime Q_do in firs inegral can be a funcion of c_v Q_do c_v d = Q_do c_v d = d => only dependen on ime c_v c_v space (locaion in c. v.) U(niform) S(ae) and second law for a uniform sae, uniform process is... uniform sae, uniform flow process Q_doc_v (7.56) = when reversible m s m s + m e s e m i s ( ) ( i ) d n n seady sae, seady flow process assumpions.... conrol volume does no move relaive o he coordinae frame. he mass in he conrol volume does no vary wih ime 3. he mass flux and he sae of mass a each discree area of flow on he conrol surface do no vary wih ime and.. he raes a which hea and work cross he conrol surface remain consan. example: cenrifugal air compressor, operaing a consan mass rae of flow, consan rae of hea ransfer o he surroundings, and consan inpu power. uniform sae, uniform flow process USUF assumpions:. conrol volume remains consan relaive o he coordinae frame. sae of mass wihin he conrol volume may change wih ime, bu a any insan of ime is uniform hroughou he enire conrol volume - I define his as f() bu no of space 3. he sae of mass crossing each of he areas of flow on he conrol surface is consan wih ime alhough he mass flow raes may be changing example: filling a closed ank wih a gas or liquid, discharge from a closed vessel. 9/5/6