ES-7A Thermodynamics HW 5: 5-62, 81, 96, 134; 7-29, 40, 42, 67, 71, 106 Spring 2003 Page 1 of 7

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1 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page of Heat Pum Given: A heat um i ued to maintain a houe at 3 C. he houe loe heat to the outide at a rate of 6, kj/h, and the houe generate 4 kj/h of heat. he COP of the heatum i.5. Find: Required ower inut to the houe. he COP of a heat um i defined a [what we want]/[what we ut in]: Q COP W H in In thi cae, the heat um need to uly enough heat to the houe to kee the houe at contant temerature: Q H Q lot Q generated 6, 4 56, kj/h. QH 56, Win,4 kj/h 6. kw. COP Carnot Heat Engine Given: A Carnot engine receive 65 kj of heat from a ource and reject kj to a ink at 7 C. Find: a) emerature of the ource, and b) thermal efficiency of the heat engine. a) For a Carnot cycle, Q L L S and Q H H S, and we can ay that: H L Q H Q H 65, o the temerature of the ource i H L K, or C. Q Q L b) he thermal efficiency of a Carnot Engine i given by: η th L / H (9/94.5).69, or 69. ercent. L 5-96 Carnot Air Conditioner Given: A revere-carnot cycle air conditioner tranfer heat from a houe at 75 kj/min to maintain it temerature at C. he outide temerature i 35 C. Find: Power required to oerate thi air conditioner. he COP for a revere-carnot air conditioner i: COP H L hi i alo equal to [what we want]/[what we ut in], or Q L /W in. W in Q L /COP 75/ kj/min.64 kw.

2 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page of Carnot Heat Engine/Refrigerator Given: A Carnot heat engine receive heat at 75 K and reject heat to 3 K. he work outut i ued to drive a Carnot refrigerator that remove heat from a -5 C ace at a rate of 4 kj/min and reject the heat to 3 K environment. Find: a) Rate of heat ulied to the heat engine, and b) total rate of heat rejection to the environment. a) he thermal efficiency of thi Carnot heat engine i: η th L / H (3/75).6 For the refrigerator, the COP i: COP H L he work inut into the refrigerator i: W in Q L /COP 4/ kj/min thi i equal to W out of the heat engine. Q in W out /η th 65./ kj/min, or.8 kw. b) he heat rejected into the environment by the Carnot Cycle i: Q out Q L Q in W in kj/min. he heat rejected by the refrigerator i: Q out Q H Q L W in kj/min. he total heat rejected i: kj/min, or 8.47 kw.

3 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page 3 of Exergy of R-34a Given: Piton/cylinder device with 5 kg of R-34a at.8 MPa and 5 C, cooled at contant reure until it exit a liquid at 3 C. Surrounding are at kpa and 3 C. Find: a) exergy of the refrigerant at initial and final tate, and b) exergy detroyed during thi roce. a) he exergy for cloed ytem, neglecting KE and PE, i: X m[u u P (v v ) ( )] At tate, we have uerheated vaor: v.846 m 3 /kg, u 6.6 kj/kg, h kj/kg,.97 kj/kgk. At tate, we have aturated liquid at 3 C: v.847 m 3 /kg, u 9.84 kj/kg, h 9.49 kj/kg,.3396 kj/kgk. At the dead tate, we have uerheated vaor: v.46 m 3 /kg, u kj/kg,. kj/kgk. X m[u u P (v v ) ( )] 5[ ( ) 33(.97.)] 4.3 kj. X m[u u P (v v ) ( )] 5[ ( ) 33(.3396.)] 3.3 kj. b) he exergy detroyed (irreveribility) i: I m ( ) Q. From the t Law, Q m(h h ) 5( ) kj (heat lo). I 5(33)( ) (-964.5) 7.8 kj. 7-4 Exergy of air Given: An inulated iton/cylinder device with 3 L of air at kpa and 7 C i heated for 5 min by a 5-W reitance heater in a contant reure roce. h urrounding are at kpa and 7 C. Find: he exergy detroyed during thi roce. Uing contant ecific heat (c.5 kj/kgk, c v.78 kj/kgk, R.87 kj/kgk): he ma of the air i: m P v /R (3/)/(.87 3).48 kg. Irreveribility i given by: I m ( ) Q P m c ln R ln ; lat term i zero becaue P P. P i found from the Firt Law: Q mc ( ) Q/mc (5 5 6/)/(.48.5) K. I.48(3)[.5 ln(657./3)] 9.88 kj

4 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page 4 of Irreveribility of Argon ga Given: An inulated rigid tank divided into two equal art by a artition. One art contain 3 kg of argon ga at 3 kpa and 7 C, and the other art i evacuated. he artition i removed, and the ga fill the entire tank. he urrounding are at 5 C. Find: Exergy detroyed during thi roce. Uing contant ecific heat (c.53 kj/kgk, c v.3 kj/kgk, R.8 kj/kgk): Since the tank i inulated, there i no heat lo during the roce: Q m c v ( ) W ince W. Irreveribility i given by: I m ( ) Q m c v ln R ln ; Q ince the tank i inulated. I m R ln 3( 98 ).8 ln 8.95 kj.

5 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page 5 of Argon ga in comreor Given: Argon ga enter an adiabatic comreor at kpa and 3 C with a velocity of m/ and leave at. MPa and 53 C, with velocity of 8 m/. he inlet area i 3 cm. he urrounding are at 5 C. Find: a) Reverible ower inut, and b) exergy detroyed. Uing contant ecific heat (c.53 kj/kgk, c v.3 kj/kgk, R.8 kj/kgk): a) he reverible ower inut i the difference in exergy between exit and inlet: W& rev X& X& Exergy for Oen Sytem i: X & m& h h ( ), o X & X& i: W& rev m& h m& c h ( ) ( ) he ma flow rate i found from the inlet: c ln v R /P.8(33)/.555 m 3 /kg m& A v ( ) kg/ P R ln P 8 83 W & rev ( 53 3) ln.8 ln 6.4 kw 33 b) he exergy detroyed, or irreveribility, i: P I& m & ln ( ) m & c ln R P ( 98).53 ln.8 ln 4. kw 33

6 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page 6 of Steam in nozzle Given: Steam enter an adiabatic nozzle at 7 MPa and 5 C with a velocity of 7 m/, and exit at 5 MPa and 45 C. he urrounding are at 5 C. Find: a) Exit velocity, b) ientroic efficiency, and c) exergy detroyed. a) he roertie of team at the entrance are: h 34.3 kj/kg, kj/kgk. he roertie at the exit are: h 336. kj/kg, kj/kgk. he exit velocity i found from the Firt Law: q h h ( h h ) ( ) 7 93,, o m/. b) o find the ientroic efficiency, we firt need to find h. At and P, we are omewhere between 4 C ( kj/kgk, h kj/kg) and 45 C ( kj/kgk, h 336. kj/kg). We interolate to find h 33.5 kj/kg. We can olve for uing h : ( h h ) ( ) 7,5, and 47.7 m/. he ientroic efficiency for a nozzle i: η / 93, /,5.868, or 86.8 ercent. c) he exergy detroyed, or irreveribility, er unit ma i: ( ) 98( ) i 6.88 kj/kg.

7 ES-7A hermodynamic HW 5: 5-6, 8, 96, 34; 7-9, 4, 4, 67, 7, 6 Sring 3 Page 7 of wo-tage team turbine Given: Steam enter an adiabatic, two-tage turbine at 8 MPa and 5 C. It exand to MPa and 35 C in the firt tage. he team i then heated at contant reure to 5 C, then enter the econd tage. At the exit of the econd tage, the team i at 3 kpa with quality of 97 ercent. he total work outut of the turbine i 5 MW, and the urrounding i at 5 C. Find: a) Reverible ower outut, and b) rate or exergy detroyed in the turbine. a) he roertie at the entrance of firt tage are: h kj/kg, 6.74 kj/kgk. he roertie at the exit of firt tage are: h 337. kj/kg, kj/kgk. he roertie at the entrance of the econd tage: h kj/kg, kj/kgk. A he roertie at the exit of the econd tage: h f 89.3 kj/kg, h fg 336. kj/kg, f.9439 kj/kgk, fg kj/kgk. h 4 h f x 4 h fg (336.) 555. kj/kg 4 f x 4 fg (6.847) kj/kgk he ma flow rate of the turbine i found from the Firt Law: Q & m& h h m& h h & ( ) ( ) W h3 ( h h h ) 5 ( ) m & W& kg/ he reverible ower outut i given by ( X & X& ) ( X& & ) &. 4 X 3 X& m[ h h ( )] [ ( 6.74) ] 4.6[ ( )] & & W rev, X kw W & kw rev, he total reverible work outut i: kw. b) he total exergy detroyed i: [( ) ( )] ( )( ) I & m & 46.7 kw. 4 3