since (Q H /T H ) = (Q L /T L ) for reversible cycles. Also, since Q diff is a positive quantity. Thus,

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1 he alidity o the Clai eqality i to be demontrated g a reerible and an irreerible heat enge operatg between the ame temperatre limit. Analyi Conider two heat enge, one reerible and one irreerible, both operatg between a hightemperatre reeroir at and a low-temperatre reeroir at. Both heat enge receie the ame amont o heat,. he reerible heat enge reject heat the amont o, and the irreerible one the amont o, irre di, where di i a poitie qantity ce the irreerible heat enge prodce le work. Notg that and are tranerred at contant temperatre o and, repectiely, the cyclic tegral o δ/ or the reerible and irreerible heat enge cycle become δ re δ δ δ ce ( / ( / or reerible cycle. Alo, δ irre,irre di δ di < 0 0 δ ce di i a poitie qantity. h, 0. Re E W net, re Irre E W net, irre, irre 7-85 he ner and er race o a wdow gla are mataed at peciied temperatre. he amont o heat traner throgh the gla and the amont o entropy eration with the gla 5 h are to be determed Amption teady operatg condition exit ce the race temperatre o the gla rema contant at the peciied ale. hermal propertie o the gla are contant. Analyi he amont o heat traner oer a period o 5 h i & cond t (. kj/( ,600 We take the gla to be the ytem, which i a cloed ytem. Under teady condition, the rate orm o the entropy balance or the gla impliie to & 44 & Rate o net entropy traner by heat and ma & b, & b, 00 W 00 W 8 K 76 K Rate o entropy eration,gla,wall 0 0 kj 0 ytem 0 44 Rate o change o entropy,gla 0.87 W/K Gla 0 C C

2 wo rigid tank that conta at dierent tate are connected by a ale. he ale i opened and team low rom tank A to tank B ntil the prere tank A drop to a peciied ale. ank B loe heat to the rrondg. he al temperatre each tank and the entropy erated drg thi proce are to be determed. Amption ank A i lated, and th heat traner i negligible. he that rema tank A ndergoe a reerible adiabatic proce. he thermal energy tored the tank themele i negligible. 4 he ytem i tationary and th ketic and potential energy change are negligible. 5 here are no work teraction. Analyi (a he team tank A ndergoe a reerible, adiabatic proce, and th. From the team table (able A-4 throgh A-6, ank A : P x P 00 kpa ( 0.8( ( 0.8( ( 0.8( ( at. mixtre, A x, A g ( ( x 56. ( ( 98. kj/kg, 400 kpa 0.8 ankb : P 00 kpa 50 C A, B, B, B, A, A x, A.989 m /kg 7.4 kj/kg kj/kg K g x x x, A, A g g he itial and the al mae tank A are and m m, A, A V A, A V A, A g , A, A 0. m kg m /kg 0. m kg m /kg g at@00 kpa.5 C 6. kj/kg kj/kg K h, kg o ma low to tank B. hen, m, B m, B kg he al peciic olme o team tank B i determed rom V A V 0. m team P 400 kpa x 0.8 ( ( ( m B kg.989 m /kg.59 m /kg B, B m, B m, B. kg m /kg m 5.9 kj/kg /kg B 600 kj m kg team 50 C P 00 kpa We take the entire content o both tank a the ytem, which i a cloed ytem. he energy balance or thi tationary cloed ytem can be expreed a E E 44 Net energy traner by heat, work, and ma Eytem 44 Change ternal, ketic, potential, etc.energie U ( U A ( U ( m m A ( m m B B (ce W KE PE 0

3 7-7 btittg, h, 600, B, B } ( 0.48( 5.9 ( 0.540( 6. } (. ( ( m /kg 5.0 kj/kg, B, B, B 5.0 kj/kg. C 7.74 kj/kg K (b he total entropy eration drg thi proce i determed by applyg the entropy balance on an extended ytem that clde both tank and their immediate rrondg o that the bondary temperatre o the extended ytem i the temperatre o the rrondg at all time. It gie 44 Net entropy traner by heat and ma b,rr Entropy eration ytem 44 Change entropy A B Rearrangg and btittg, the total entropy erated drg thi proce i determed to be A B, B ( m m ( m m ( 0.48( ( 0.540( } (.( 7.74 ( ( } 0.96 kj/k b,rr A B b,rr 600 kj 7 K 7-87 eat i tranerred teadily to boilg a pan throgh it bottom. he rate o entropy eration with the bottom plate i to be determed. Amption teady operatg condition exit ce the race temperatre o the pan rema contant at the peciied ale. Analyi We take the bottom o the pan to be the ytem, which i a cloed ytem. Under teady condition, the rate orm o the entropy balance or thi ytem can be expreed a & 44 & Rate o net entropy traner by heat and ma & b, & b, 500 W 500 W 78 K 77 K Rate o entropy eration,ytem,ytem 0 ytem Rate o change o entropy,ytem W/K 500 W 04 C 05 C Dicion Note that there i a mall temperatre drop acro the bottom o the pan, and th a mall amont o entropy eration.

4 An electric reitance heater i immered. he time it will take or the electric heater to raie the temperatre to a peciied temperatre and the entropy erated drg thi proce are to be determed. Amption Water i an compreible btance with contant peciic heat. he energy tored the contaer itel and the heater i negligible. eat lo rom the contaer i negligible. Propertie he peciic heat o at room temperatre i c 4.8 kj/kg C (able A-. Analyi akg the the contaer a the ytem, which i a cloed ytem, the energy balance can be expreed a btittg, E E 44 Net energy traner by heat, work, and ma W& W e, e, Change ternal, ketic, potential, etc.energie ( U t mc( Eytem 44 (00 J/ t (40 kg(480 J/kg C(50-0 C olg or t gie t m.6 h Water 40 kg eater Aga we take the the tank to be the ytem. Notg that no heat or ma croe the bondarie o thi ytem and the energy and entropy content o the heater are negligible, the entropy balance or it can be expreed a 44 Net entropy traner by heat and ma Entropy eration 0 ytem 44 Change entropy hereore, the entropy erated drg thi proce i mc ln K ( 40 kg( 4.8 kj/kg K ln 6. kj/k 9 K 7-89 A hot pipe at a peciied temperatre i log heat to the rrondg air at a peciied rate. he rate o entropy eration the rrondg air de to thi heat traner are to be determed. Amption teady operatg condition exit. Analyi We take the air the icity o the pipe (excldg the pipe a or ytem, which i a cloed ytem.. he ytem extend rom the er race o the pipe to a ditance at which the temperatre drop to the rrondg temperatre. In teady operation, the rate orm o the entropy balance or thi ytem can be expreed a & 44 & Rate o net entropy traner by heat and ma & b, & b, 00 W 00 W 5 K 78 K Rate o entropy eration,ytem,ytem 0 ytem Rate o change o entropy,ytem.68 W/K 80 C Air, 5 C

5 he eed o a team power plant i preheated g team extracted rom the trbe. he ratio o the ma low rate o the extracted team to the eed and entropy eration per nit ma o eed are to be determed. Amption hi i a teady-low proce ce there i no change with time. Ketic and potential energy change are negligible. eat lo rom the deice to the rrondg i negligible. Propertie he propertie o team and eed are (able A-4 throgh A-6 P MPa h 88. kj/kg 00 C kj/kg K h h P MPa at. liqid P.5 MPa h h 50 MPa 76.5 kj/kg.8 kj/kg K C C C P4.5 MPa h4 h o o 4 0 C 70 C kj/kg kj/kg K C C kj/kg.047 kj/kg K team rom trbe Analyi (a We take the heat exchanger a the ytem, which i a control olme. he ma and energy balance or thi teady-low ytem can be expreed the rate orm a ollow: Ma balance (or each lid tream: 4 MPa 00 C at. liqid 0 (teady ytem 0 and 4 w Energy balance (or the heat exchanger: E E & 44 & Rate o net energy traner by heat, work, and ma E& h h Rate o change ternal, ketic, potential, etc.energie E& h 0 (teady Eytem & 4444 h Combg the two, ( h h ( h Diidg by 4 4 w h4 0 (ce & W& ke pe 0 h4 h ( &m w and btittg, h h ( w kj/kg 0.47 kj/kg Feed.5 MPa (b he total entropy change (or entropy eration drg thi proce per nit ma o eed can be determed rom an entropy balance expreed the rate orm a & 44 & Rate o net entropy traner by heat and ma ( w w ( 4 4 Rate o entropy eration 4 4 w 0 ytem Rate o change o entropy ( ( ( 0.47( ( 0. kj/k per kg o eed

6 EE Problem 7-90 i reconidered. he eect o the tate o the team at the let to the eed heater i to be etigated. he entropy o the extraction team i amed to be contant at the ale or MPa, 00 C, and the extraction team prere i to be aried rom MPa to 00 kpa. Both the ratio o the ma low rate o the extracted team and the eed heater and the total entropy change or thi proce per nit ma o the eed are to be plotted a nction o the extraction prere. Analyi he problem i oled g EE, and the relt are tablated and plotted below. "Known:" WorkFlid$ 'team_iapw' "P[] 000 [kpa]" "place } arond P[] and [] eqation to ole the table" [] 00 [C] P[4] P[] x[4]0 [4]temperatre(WorkFlid$,PP[4],xx[4] P[] 500 [kpa] [] 50 [C] P[] 500 [kpa] [] [4] - 0"[C]" "ce we don't know the ma low rate and we want to determe the ratio o ma low rate o the extracted team and the eed, we can ame the ma low rate o the eed i kg/ with lo o erality. We write the coneration o energy." "Coneration o ma or the team extracted rom the trbe: " m_dot_team[] m_dot_team[4] "Coneration o ma or the condenate lowg throgh the eed heater:" m_dot_w[] m_dot_w[] m_dot_w[] "Coneration o Energy - F energy balance or the eed heater -- neglectg the change potential energy, no heat traner, no work:" h[]enthalpy(workflid$,pp[],[] "o ole the table, place } arond [] and remoe them rom the nd and rd eqation" []entropy(workflid$,pp[],[] [] 6.69 [kj/kg-k] "hi [] i or the itial [], P[]" []temperatre(workflid$,pp[],[] "Ue thi eqation or [] only when [] i gien."} h[4]enthalpy(workflid$,pp[4],xx[4] [4]entropy(WorkFlid$,PP[4],xx[4] h[]enthalpy(workflid$,pp[],[] []entropy(workflid$,pp[],[] h[]enthalpy(workflid$,pp[],[] []entropy(workflid$,pp[],[] "For the eed heater:" E_dot_ E_dot_ E_dot_ m_dot_team[]*h[] m_dot_w[]*h[] E_dot_ m_dot_team[4]*h[4] m_dot_w[]*h[] m_ratio m_dot_team[]/ m_dot_w[] "econd aw analyi:" _dot_ - _dot dot_ DEA_dot_y DEA_dot_y 0 "[KW/K]" "teady-low relt" _dot_ m_dot_team[]*[] m_dot_w[]*[] _dot_ m_dot_team[4]*[4] m_dot_w[]*[] PerUnitMaFW _dot_/m_dot_w[]"[kj/kg_w-k]"

7 7-4 m ratio,perunitma [kj/kg-k] P [kpa] ] g wḵ k J / [k W F a M n it P e ru n, ge P 000 kpa 0. For P < 7 kpa 0.9 < P 7 kpa m ratio

8 E A rigid tank itially conta atrated R-4a apor. he tank i connected to a pply le, and i charged ntil the tank conta atrated liqid at a peciied prere. he ma o R-4a that entered the tank, the heat traner with the rrondg at 0 F, and the entropy erated drg thi proce are to be determed. Amption hi i an nteady proce ce the condition with the deice are changg drg the proce, bt it can be analyzed a a niorm-low proce ce the tate o lid at the let rema contant. Ketic and potential energie are negligible. here are no work teraction oled. 4 he direction o heat traner i to the tank (will be eriied. Propertie he propertie o R-4a are (able A- throgh A- P 00 pia at. apor P 0 pia at. liqid g@00 pia g@00 pia g@00 pia t Bt/lbm 0.98 Bt/lbm R t /lbm /lbm 4.49 Bt/lbm Bt/lbm R R-4a 0 F 40 pia 80 F R-4a t Pi 40 pia hi F 8.7 Bt/lbm i 80 F F Bt/lbm R Analyi (a We take the tank a the ytem, which i a control olme ce ma croe the bondary. Notg that the energie o lowg and nonlowg lid are repreented by enthalpy h and ternal energy, repectiely, the ma and energy balance or thi niorm-low ytem can be expreed a Ma balance: m m mytem mi m m Energy balance: E E E 44 ytem 44 Net energy traner by heat, work, and ma Change ternal, ketic, potential, etc.energie mihi m m he itial and the al mae the tank are (ce W ke pe 0 V t V t m 6.8 lbm m 0.55 lbm t /lbm t /lbm hen rom the ma balance, m i m m lbm (b he heat traner drg thi proce i determed rom the energy balance to be m h m m i i ( 4. lbm( 8.7 Bt/lbm ( 0.55 lbm( 4.49 Bt/lbm ( 6.8 lbm( Bt/lbm Bt (c he entropy erated drg thi proce i determed by applyg the entropy balance on an extended ytem that clde the tank and it immediate rrondg o that the bondary temperatre o the extended ytem i the temperatre o the rrondg at all time. he entropy balance or it can be expreed a 44 Net entropy traner by heat and ma Entropy eration ytem 44 Change entropy hereore, the total entropy erated drg thi proce i mi i ( m m b, b, m i i tank m Bt m ( 4.( ( 0.55( ( 6.8( Bt/R 570 R

9 It i to be hown that or thermal energy reeroir, the entropy change relation mc ln( / redce to / a. Analyi Conider a thermal energy reeroir o ma m, peciic heat c, and itial temperatre. Now heat, the amont o, i tranerred to thi reeroir. he irt law and the entropy change relation or thi reeroir can be written a and mc ( mc ln ln ( / mc akg the limit a by applyg the 'opital' rle, / which i the deired relt. hermal energy reeroir m, c, 7-94 he ner and er glae o a doble pane wdow are at peciied temperatre. he rate o entropy traner throgh both ide o the wdow and the rate o entropy eration with the wdow are to be determed. Amption teady operatg condition exit ce the race temperatre o the gla rema contant at the peciied ale. Analyi he entropy low aociated with heat traner throgh the let and right glae are let right & & let let right right 0 W K 0 W K W/K W/K We take the doble pane wdow a the ytem, which i a cloed ytem. In teady operation, the rate orm o the entropy balance or thi ytem can be expreed a & 44 & Rate o net entropy traner by heat and ma & b, & b, 0 W 0 W 9 K 79 K Rate o entropy eration,ytem,ytem 0 ytem Rate o change o entropy,ytem 0.06 W/K 8 C Air 6 C

10 A well-lated room i heated by a team radiator, and the warm air i ditribted by a an. he aerage temperatre the room ater 0 m, the entropy change o team and air, and the entropy erated drg thi proce are to be determed. Amption Air i an ideal ga with contant peciic heat at room temperatre. he ketic and potential energy change are negligible. he air prere the room rema contant and th the air expand a it i heated, and ome warm air ecape. Propertie he ga contant o air i R 0.87 kpa.m /kg.k (able A-. Alo, c p.005 kj/kg.k or air at room temperatre (able A-. Analyi We irt take the radiator a the ytem. hi i a cloed ytem ce no ma enter or leae. he energy balance or thi cloed ytem can be expreed a E E 44 Net energy traner by heat, work, and ma Change ternal, ketic, potential, etc.energie U m( m( Eytem 44 (ce W KE PE 0 Ug data rom the team table (able A-4 throgh A-6, ome propertie are determed to be P 00 kpa.0805 m /kg kj/kg 00 C kj/kg.k P ( x btittg, 00 kpa , 47.40,.08, V 0.05 m m kg.0805 m /kg g x x g.694 m /kg 088. kj/kg g g g g kj/kg.k kj/kg kj/kg.k (0.088 kg( kJ/kg.6 kj he olme and the ma o the air the room are V m³ and m ( 00 kpa P V (80 m air R (0.870 kpa m /kg K he amont o an work done 0 m i W W& t (0.0 kj/(0 60 ( 8 K an, an, 6kJ 98.5 kg 0 C 4 m 4 m 5 m team radiator We now take the air the room a the ytem. he energy balance or thi cloed ytem i expreed a W an, E E W W b, an, E U ytem mc p (

11 7-45 ce the bondary work and U combe to or a contant prere expanion or compreion proce. btittg, which yield (.6 kj (6 kj (98.5 kg(.005 kj/kg C( - 0 C. C hereore, the air temperatre the room rie rom 0 C to. C 0 m. (b he entropy change o the team i team m ( ( kg( kj/kg K 0.05 kj/k (c Notg that air expand at contant prere, the entropy change o the air the room i mc air p ln P mr ln P K ( 98.5 kg(.005 kj/kg K ln 0.80 kj/k 8 K (d We take the air the room (cldg the team radiator a or ytem, which i a cloed ytem. Notg that no heat or ma croe the bondarie o thi ytem, the entropy balance or it can be expreed a 44 Net entropy traner by heat and ma Entropy eration ytem 44 Change entropy 0 team btittg, the entropy erated drg thi proce i determed to be team air air kj/k

12 he heatg o a paie olar hoe at night i to be aited by olar heated. he length o time that the electric heatg ytem wold rn that night and the amont o entropy erated that night are to be determed. Amption Water i an compreible btance with contant peciic heat. he energy tored the gla contaer themele i negligible relatie to the energy tored. he hoe i mataed at C at all time. Propertie he denity and peciic heat o at room temperatre are ρ kg/ and c 4.8 kj/kg C (able A-. Analyi he total ma o i m w ρv ( kg/( kg akg the content o the hoe, cldg the a or ytem, the energy balance relation can be written a or, E 44 Net energy traner by heat, work, and ma W& btittg, It gie W E e, Change ternal, ketic, potential, etc.energie U ( U ( U mc( Eytem 44 e, t [ mc( ] ( U (5 kj/ t - (50,000 kj/h(0 h (000 kg(4.8 kj/kg C( - 80 C t 7, h air 80 C C 50,000 kj/h We take the hoe a the ytem, which i a cloed ytem. he entropy erated drg thi proce i determed by applyg the entropy balance on an extended ytem that clde the hoe and it immediate rrondg o that the bondary temperatre o the extended ytem i the temperatre o the rrondg at all time. he entropy balance or the extended ytem can be expreed a 44 Net entropy traner by heat and ma b, Entropy eration ytem 44 Change entropy 0 air ce the tate o air the hoe rema nchanged. hen the entropy erated drg the 0-h period that night i b, ( 000 kg( 4.8 kj/kg kj/k mc ln rr 95 K 500,000 kj K ln 5 K 76 K

13 E A teel contaer that i illed with hot i allowed to cool to the ambient temperatre. he total entropy erated drg thi proce i to be determed. Amption Both the and the teel tank are compreible btance with contant peciic heat at room temperatre. he ytem i tationary and th the ketic and potential energy change are zero. peciic heat o iron can be ed or teel. 4 here are no work teraction oled. Propertie he peciic heat o and the iron at room temperatre are c p,.00 Bt/lbm. F C p, iron 0.07 Bt/lbm. C. he denity o at room temperatre i 6. lbm/t³ (able A-E. Analyi he ma o the i m ρvv (6. lbm/t (5 t 9.5 lbm We take the teel contaer and the it a the ytem, which i a cloed ytem. he energy balance on the ytem can be expreed a E E 44 Net energy traner by heat, work, and ma Eytem 44 Change ternal, ketic, potential, etc.energie U U contaer contaer U [ mc( ] [ mc( ] btittg, the heat lo to the rrondg air i determed to be [ mc( ] contaer [ mc( ] (75 lbm(0.07 Bt/lbm 46,976 Bt o WAER 0 F teel 70 F F(0 70 F (9.5 lbm(.00 Bt/lbm F(0 70 F We aga take the contaer and the In it a the ytem. he entropy erated drg thi proce i determed by applyg the entropy balance on an extended ytem that clde the contaer and it immediate rrondg o that the bondary temperatre o the extended ytem i the temperatre o the rrondg air at all time. he entropy balance or the extended ytem can be expreed a where 44 Net entropy traner by heat and ma contaer b, mc mc Entropy eration ag ag ln ln ytem 44 Change entropy contaer ( 75 lbm( 0.07 Bt/lbm R hereore, the total entropy erated drg thi proce i contaer 50 R ln 0.7 Bt/ R 580 R 50 R ( 9.5 lbm(.00 Bt/lbm R ln 8.98 Bt/ R b, 580 R 46,976 Bt R Bt/R and

8-4 P 2. = 12 kw. AIR T = const. Therefore, Q &

8-4 P 2. = 12 kw. AIR T = const. Therefore, Q & 8-4 8-4 Air i compreed teadily by a compreor. e air temperature i mataed contant by eat rejection to te urroundg. e rate o entropy cange o air i to be determed. Aumption i i a teady-low proce ce tere i