(b) The heat transfer can be determined from an energy balance on the system

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Cameron Lynch
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1 8-5 Heat is transferred to a iston-cylinder device wit a set of stos. e work done, te eat transfer, te exergy destroyed, and te second-law efficiency are to be deterined. Assutions e device is stationary and kinetic and otential energy canges are zero. ere is no friction between te iston and te cylinder. Analysis (a) e roerties of te refrigerant at te initial and final states are (ables A- troug A-) ka.65 /kg v C 8. kj/kg u s.6 kj/kg.k 8 ka.756 /kg v C.96 kj/kg u s.8 kj/kg.k e boundary work is deterined to be W ( ) (. kg)(8 ka)( ) b, out v v /kg.57 kj (b) e eat transfer can be deterined fro an energy balance on te syste in ( u u) + Wb, out (. kg)(.96 8.)kJ/kg +.57 kj 9.8 kj (c) e exergy difference between te inlet and exit states is X u [ u ( s s ) + ( v v )] (. kg).6 kj [(.96 8.)kJ/kg (98 K)(.8.6)kg.K + ( ka)( ) /kg] e useful work outut for te rocess is W u, out Wb,out ( v v).57 kj (. kg)( ka)( ) /kg. kj e exergy destroyed is te difference between te exergy difference and te useful work outut X X.6..7 kj dest W u,out (d) e second-law efficiency for tis rocess is Wu,out η II X. kj.6 kj.78 -a. kg ka C

2 8-9 Stea exands in a turbine, wic is not insulated. e reversible ower, te exergy destroyed, te second-law efficiency, and te ossible increase in te turbine ower if te turbine is well insulated are to be deterined. Assutions Steady oerating conditions exist. otential energy cange is negligible. Analysis (a) e roerties of te stea at te inlet and exit of te turbine are (ables A- troug A-6) a 8.7 kj/kg 55 C s kj/kg.k ka 9. kj/kg x.95 s kj/kg.k e entaly at te dead state is 5 C.8 kj/kg x e ass flow rate of stea ay be deterined fro an energy balance on te turbine V V & + & + + & out + a (6 /s) kj/kg ( /s) & 8.7 kj/kg + 9. kj/kg & + /s + 5 kw + 5 kw &.69 kg/s e reversible ower ay be deterined fro V -V W & rev & ( s s ) + (6 /s) (.69) (8.7 9.) (98)( ) + 7kW ( /s) kj/kg /s kj/kg /s (b) e exergy destroyed in te turbine is X & dest rev a kW (c) e second-law efficiency is a 5 kw η II.7 rev 7 kw (d) e energy of te stea at te turbine inlet in te given dead state is & & ( ) (.69 kg/s)( )kJ/kg 995 kw e fraction of energy at te turbine inlet tat is converted to ower is a 5 kw f.79 & 995 kw Assuing tat te sae fraction of eat loss fro te turbine could ave been converted to work, te ossible increase in te ower if te turbine is to be well-insulated becoes W & f & (.79)(5 kw). kw increase out Stea a 55 C, 6 /s urbine ka /s x.95

3 8-6 Argon gas is exanded adiabatically in an exansion valve. e exergy of argon at te inlet, te exergy destruction, and te second-law efficiency are to be deterined. Assutions Steady oerating conditions exist. Kinetic and otential energy canges are zero. Argon is an ideal gas wit constant secific eats. roerties e roerties of argon gas are.8 kj/kg.k, c.5 kj/kg.ºc (able A-). Analysis (a) e exergy of te argon at te inlet is x c ( ( s s ) ) c ln ln (.5 kj/kg.k)( 5) C (98 K) (.5 kj/kg.k)ln.7 kj/kg 7 K 98 K (.8 kj/kg.k)ln (b) oting tat te teerature reains constant in a trottling rocess of an ideal gas, te exergy destruction is deterined fro x dest s ( s gen s ) ln (98 K) (.8 kj/kg.k)ln.7 kj/kg (c) e second-law efficiency is x η II x x dest (.7.7)kJ/kg.6.7 kj/kg Argon.5 a C 5 ka 5 ka 5 ka 5 ka ka

4 -9 A geoteral eat u is considered. e degrees of subcooling done on te refrigerant in te condenser, te ass flow rate of te refrigerant, te eating load, te of te eat u, te iniu ower inut are to be deterined. Assutions Steady oerating conditions exist. Kinetic and otential energy canges are negligible. Analysis (a) Fro te refrigerant-a tables. (ables A- troug A-) x C. & 57. ka. kj/kg in ( out H & ( ) (.96 kg/s)(8..) kj/kg.7 kw & H.7 kw.68 in.656 kw (d) e reversible of te cycle is rev.9 L / H (5 + 7) /(5 + 7) e corresonding iniu ower inut is &.7 kw H in,in.8 kw.9 rev H Condenser Exansion 57. ka 6.59 kj/kg valve Coressor x (sat. va.) s.9 kj/kg ka 8. kj/kg s s Evaorator C sat. va. Fro te stea tables (able A-) x. w 5 C 9. kj/kg C Water w C 67.5 kj/kg 5 C e saturation teerature at te condenser ressure of ka and te actual teerature at te condenser outlet are ka 5. C H. a ka 8.59 C (fro EES). kj/kg en, te degrees of subcooling is subcool sat C s (b) e rate of eat absorbed fro te L geoteral water in te evaorator is & L & w ( w w ) (.65 kg/s)( )kJ/kg.78 kw is eat is absorbed by te refrigerant in te evaorator & L.78 kw &.96 kg/s (6.59.)kJ/kg (c) e ower inut to te coressor, te eating load and te are & ) + & (.96 kg/s)( )kJ/kg.656 kw. a s s W in s. W in

5 -66 A gas refrigeration cycle wit eliu as te working fluid is considered. e iniu teerature in te cycle, te, and te ass flow rate of te eliu are to be deterined. Assutions Steady oerating conditions exist. Heliu is an ideal gas wit constant secific eats. Kinetic and otential energy canges are negligible. roerties e roerties of eliu are c 5.96 kj/kgk and k.667 (able A-). Analysis (a) Fro te isentroic relations, s ( k ) / k H.667 /.667 s ( 6K)( ) 8.K 5 C ( k ) / k.667 / C s ( K) 8.K and s efrig s η η ( s ) (.8)( 8.) s s. K ηc s s + s.5 K (b) e of tis gas refrigeration cycle is deterined fro q w net,in w w ( ) ( ) ( ) ( ) L co,in 6. (.5 6) (.) q L turb,out in ( )/ η 6 + ( 8. 6) /(.8).56 (c) e ass flow rate of eliu is deterined fro & refrig & refrig & refrig 8 kj/s &.9 kg/s q c 5.96 kj/kg K 6. K L ( ) ( )( ) C

6 -8E A two-evaorator coression refrigeration cycle wit refrigerant-a as te working fluid is considered. e cooling load of bot evaorators er unit of flow troug te coressor and te of te syste are to be deterined. Assutions Steady oerating conditions exist. Kinetic and otential energy canges are negligible. Coressor Exansion Exansion valve valve Evaorator 5 Exansion valve 7 Evaorator Analysis Fro te refrigerant tables (ables A-E, A-E, and A-E), 6 6 sia 6 sia 8.59 Btu/lb sat. liquid 6 5 F sat. vaor 8.59 Btu/lb 5 Condenser F (trottling) 7. Btu/lb F F Btu/lb sat. vaor For a unit ass flowing troug te coressor, te fraction of ass flowing troug Evaorator II is denoted by x and tat troug Evaorator I is y (y -x). Fro te cooling loads secification, & L,e va & L,e va x( 5 ) y( 7 6 ) were x y Cobining tese results and solving for y gives y.698 ( 7 6 ) + ( 5 ) ( ) + ( ) en, x y Alying an energy balance to te oint in te syste were te two evaorator streas are recobined gives x5 + y7 (.6)(7.) + (.698)(98.68) x 5 + y7.8 Btu/lb en, 6 sia 6 F H -9.5 F 5 L 7 W in s

7 9.5 F.8 Btu/lb s 6 sia s sia s. Btu/lb.8 Btu/lb e cooling load of bot evaorators er unit ass troug te coressor is q L x( 5 ) + y( 7 6 ) (.6)( ) Btu/lb + (.698)( ) Btu/lb Btu/lb e work inut to te coressor is w in (..8) Btu/lb 6.96 Btu/lb e of tis refrigeration syste is deterined fro its definition, q L w in Btu/lb 6.96 Btu/lb.6 - e ass fractions of te constituents of a gas ixture are given. e ole fractions of te gas and gas constant are to be deterined. roerties e olar asses of, and are 6. and. kg/kol, resectively (able A-) Analysis For convenience, consider kg of te ixture. en te nuber of oles of eac coonent and te total nuber of oles are 75 kg 5 kg kol.568 kol + en te ole fraction of eac coonent becoes y y.688 kol.89 or 89.% 5.56 kol.568 kol.8 or.8% 5.56 kol 75 kg.688 kol 6 kg/kol 5 kg.568 kol kg/kol 5.56 kol e olar ass and te gas constant of te ixture are deterined fro teir definitions, and u kg 9. kg/ kol 5.56 kol 8. kj/kol K.7 kj/kg K 9. kg/kol ass 75% 5% - e asses, teeratures, and ressures of two gases contained in two tanks connected to eac oter are given. e valve connecting te tanks is oened and te final teerature is easured. e volue of eac tank and te final ressure are to be deterined.

8 Assutions Under secified conditions bot and can be treated as ideal gases, and te ixture as an ideal gas ixture roerties e olar asses of and are 8. and. kg/kol, resectively. e gas constants of and are.968 and.598 ka /kgk, resectively (able A-). Analysis e volues of te tanks are V V ( kg)(.968 ka /kg K)(98 K).95 ka ( kg)(.598 ka /kg K)(98 K).65 5 ka kg 5 C ka kg 5 C 5 ka Also, us, V total V + V.95 kg 8 kg/kol +.65 kg.975 kol kg/kol.57 kol kol kol u V (.95 kol)(8. ka kol /kol K)(98 K). ka

9 -7 Heat is transferred to a gas ixture contained in a iston cylinder device. e initial state and te final teerature are given. e eat transfer is to be deterined for te ideal gas and non-ideal gas cases. roerties e olar asses of H and are., and 8. kg/kol. (able A-). Analysis Fro te energy balance relation, E E E in in W out b,out in U H H + H ( ) H + ( ) H H since W b and U cobine into H for quasi-equilibriu constant ressure rocesses H H 6kg kol H kg / kol kg.75 kol 8 kg / kol (a) Assuing ideal gas beavior, te inlet and exit entalies of H and are deterined fro te ideal gas tables to be K,55. kj / K 5,669. kj / kol :,68 kj / kol, 5,8 kj / K us, ideal ( 5,669.,55.) +.75 ( 5,8, 68) 7 kj (b) Using Aagat's law and te generalized entaly dearture cart, te entaly cange of eac gas is deterined to be 6,.85,H cr,h. Z 5 H :.86,H,H cr,h. Z, 6.6,H cr,h. us H can be treated as an ideal gas during tis rocess. :, erefore,,,, cr, cr,,, ( ) ( ) 6 6. cr, ( ) ( Z Z ) + ( ) H u cr H,ideal Z.7 Z..7 5,669.,55.,.8kJ/kol ideal (Fig. A-9) (Fig. A-9) (8.ka /kol K)(6.K)(..7) + (5,8,68)kJ/kol,79.5kJ/kol in kol,.8 kj/kol +.75 kol,79.5 kj/kol Substituting, ( )( ) ( )( ) 75 kj 6 kg H kg 5 a 6 K

10 - e asses of coonents of a gas ixture are given. is ixture is exanded in an adiabatic, steady-flow turbine of secified isentroic efficiency. e second law efficiency and te exergy destruction during tis exansion rocess are to be deterined. Assutions All gases will be odeled as ideal gases wit constant secific eats. roerties e olar asses of,, and He are.,., and. kg/kol, resectively (able A-). e constant-ressure secific eats of tese gases at roo teerature are.98,.86, and 5.96 kj/kg K, resectively (able A-a). Analysis e total ass of te ixture is + + He kg ka 7 C e ole nubers of eac coonent are.kg.5 kol kg/kol kg.7 kol kg/kol He.5 kg He.5 kol He kg/kol e ole nuber of te ixture is + + He e aarent olecular weigt of te ixture is.6 kg.6 kg/kol.586 kol e aarent gas constant of te ixture is u 8. kj/kol K.786 kj/kg K.6 kg/kol e ass fractions are f f f He He.kg.65.6 kg kg.6 kg.65.5 kg.5.6 kg e constant-ressure secific eat of te ixture is deterined fro c f c, + f c, + f Hec kj/kg K en te constant-volue secific eat is c v, He c kj/kg K e secific eat ratio is c.9 k.55 c v.5 e teerature at te end of te exansion for te isentroic rocess is.586 kol,, He ixture ka

11 ( k ) / k.55/.55 ka s (6 K) 65 K ka Using te definition of turbine isentroic efficiency, te actual outlet teerature is ηturb ( s ) (6 K) (.9)(6 65) e entroy cange of te gas ixture is 99 K 99 s c ln ln (.9) ln (.786) ln.658 kj/kg K 6 s e actual work roduced is wout c ( ) (.9 kj/kg K)(6 99) K 665 kj/kg e reversible work outut is w rev, out ( s s ) 665 kj/kg (98 K)(.658 kj/kg K) 7 kj/kg e second-law efficiency and te exergy destruction are ten η x II dest w w out rev,out w w kj/kg rev,out out

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