(b) Using the ideal gas equation of state, and noting that the total mass of gas occupies the same total volume at the final state as initially: where

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1 6.55 Given: An inulated cylinder i initially divided int halve y a itn. On either ide the itn i a ga at a knwn tate. The itn i releaed and equiliriu i attained. Find: Deterine the inal reure, inal teerature, and the aunt entry rduced. Autin: () The yte cnit th quantitie ga and the itn. But the itn exerience n net change in tate. () The itn ve reely in the cylinder and i therally cnducting. (3) The ga i deled a an ideal ga. (4) =0. Analyi: (a) An energy alance reduce t give U W U 0 itn U U U 0 [ u ( T ) u 300K ] [ u ( T ) u 300 K ] 0 u ( T ) u 300K T 300K () Uing the ideal ga equatin tate, and nting that the ttal a ga ccuie the ae ttal vlue at the inal tate a initially: where Thu, / / RT RT /, 3, RT RT RT RT / RT ince T T T / / / RT RT RT = =.5 ar

2 (c) An entry alance reduce t give S T r S S S S itn where S [ ( T ) ( T ) R ln ] R ln S [ ( T ) ( T ) R ln ] R ln Thu / / R[ ln ln ] R[ ln ln ] RT RT / [ ln ln ] T = kN / kj = [ 00ar ln + ar ln ] 300K ar kn = = kj/k 6.63 Given: A ga lw thrugh a ne inlet, ne exit cntrl vlue erating at teady tate. Heat traner take lace nly at teerature T. Find: Fr each everal cae, deterine i the change in eciic entry r inlet t exit i itive, negative, r indeterinate. Autin: () The cntrl vlue hwn in the accanying igure erate at teady tate. () Heat traner ccur nly where the teerature i T.

3 Analyi: At teady-tate Eq.6.4 reduce t Eq Then, ince heat traner take lace at T and there i ingle inlet and ingle exit, the entry rate alance ece 0 ( ) T Slving r the change in eciic entry r inlet t exit T (a) =0, =0 0 T () =0, <0 0 T (c) =0, >0 0 T (d) >0, <0 T (e) >0, 0 0 T 6.68 Given: H ga enter an inulated cntrl vlue erating at teady tate at 35 C and reure. Hal exit at 90 C, ar and hal exit at -0 C, ar. Fr the cntrl vlue, W =0. Find: Deterine the iniu allwed value r.

4 Autin: () The cntrl vlue hwn in the accanying igure erate at teady tate. () The eect kinetic and tential energy are negligile. (3) H i deled a an ideal ga with cntant c. Analyi: The ga inratin ut atiy the cnervatin energy rincile. Thu, with W 0 and autin and an energy rate alance reduce t give 0 W h h h 3 3 Or, with / 3 0 h h h 3 A c i cntant, h c ( T T ) where T i an aritrary reerence re re teerature. With thi r h, h and h the energy rate alance give 3 Intrducing the given teerature 0 T T T Thu, the cnervatin energy rincile i atiied. Turning next t the ecnd law, an entry rate alance reduce t give T j With / 3 0 ( ) ( ) 3 Intrducing Eq.6.3 with = 3 and =

5 T T T T 0 [ c ln R ln ] [ c ln R ln ] 3 c T ln ln( ) ( ) R T T R 3 = 4.3kJ/kg K 308 σ [ ]ln + ( ) ( ) 8.34/.06 kj/kg K R R Since = ar,.ar ex ( ) R where >0. Thu, in the liit a 0,. ar. 6.7 Given: Air enter an inulated diuer erating at teady tate at ar, -3 C, 60 / and exit at 30 /. Find: Deterine (a) the teerature the air at the exit, () the axiu attainale exit reure. Autin: () The diuer erate at teady tate and i inulated. () The change in tential energy r inlet t exit can e neglected. (3) Air i deled a an ideal ga. Analyi: At teady tate and with tated autin an energy rate alance reduce t 0 W [ h h g z z ]

6 Or h h With h r Tale A- at 70 K h N kj 70.+ kg / 0 N 3 =95.46 kj/kg Tale A- then give T =95 K An entry rate alance reduce t read Since 0,. Fr ixed exit teerature T ( and thu ixed exit velcity ), the accanying T- diagra hw that the axiu allwed exit reure crrend t the cae. T deterine thi reure,, ue Eq.6.a t write 0 ( T ) ( T ) R ln with T T T T ln ( ) ( ) R With data r Tale A- at T =70 K, T =95 K ln =.363 ar 8.34 / 8.97