Thermodynamics 1 ideal gas PDF

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The compression ratio of Otto cycle is 10. If the maximum temperature of the cycle is 1200C, find the temperature after expansion. rk= 10v 1 v 2=v 3 v 4=rkt 3 t 4=¿ 1200 + 273t 4= 10 1− 1 t 4 =586.4119K=313 The initial temperature and pressure of Otto cycle are 28C and 100kpa respectively. The compr...

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50. The compression ratio of Otto cycle is 10. If the maximum temperature of the cycle is 1200C, find the temperature after expansion.

v4 v3 ¿ ¿ t3 =¿ t4

v1 v 3 = =rk v2 v 4

rk =10

1200 + 273 =101.4−1 t4

t 4=586.4119 K =313.41 C

51. The initial temperature and pressure of Otto cycle are 28C and 100kpa respectively. The compression ratio is 6 and heat added during the process is 1300KJ/kg. Find the maximum Pressure of the cycle.

rk =6

k−1 ¿ ¿ rk t2 =¿ t1

@1 ¿ 2

6 ¿ ¿ t 2=28+273 ¿ @2 ¿ 3

1.4

¿ 100 ( 6) =1228.6035 kpa

P3 T 3 = P2 T 2

(

P3=1228.60 2158.99+

¿ 616.3494 K=343.3494 C

1300=( 0.716 )( T 3−343.35 )

QA=mcV ∆ T

)

273 +273 343.35

k

P 2= P 1 ( rk )

T 3=2158.9919C

¿ 4847.8244 KpA

52. An ocean thermal energy conversion power plant generates 10,000 kw using a warm surface water inlet temperature of 26C and a cold deep water temperature of 15C. on the basis of a 3C drop in the temperature of warm water and a 3C rise in the temperature of the cold water due to removal and addition of heat, calculate the power required in kw to pump the cold deep water to the surface and through the system heat exchanger if the required pumping pressure increase is 7kPa. Assume Carnot cycle efficiency and density of cold water to be 1000kg/m3.

¿ 1−

W =1000 kw ¿ 261818.13 kw

¿ 20.843

m3 s

TL TH

¿ 1−

15 + 273 26 + 273

¿ 7 ( 20.843 )

¿ 145.906 kw

W −w e 20843.73711 kg m= s

QR=QA −W

261818.18=m ( 4.187 )( 3 )

QR=mCP ∆T

w= pv

¿ 0.0367893

¿

10000 −1000 0.0367 m 20843.73 v= = 1000 p ¿

¿ 146 kw

53. An elastic sphere containing gas at 120kPa has a diameter of 1.0m. Heating the sphere causes it to expand to a diameter of 1.2m. During process the pressure is proportional to the sphere diameter. Calculate the work done by the gas in KJ. P= kD

¿k 2r ¿

120=k ( 1)

960 4 πr 4

¿ 240 πr

k =120 4

p=k 2 r

¿ 240 r

Pdv ¿ w=∫ ¿

( )

240 r ( 4 π ) r 2 dr ¿ ∫¿

∫ 960 π r 3 dr

54. An ideal gas with a molecular weight of 7.1 kg/kgmol is compressed from 600kpa and 280K to a final specific volume of 0.5 m3/kg. During the process the pressure varies according to P=600 + 150v + 95v2 where p is in kpa and v in m3/kg. Calculate the work of compression In KJ/kg?

(

600 V 1=

p1 v 1=RT 1

)

8.314 (280 ) 7.1

¿ ( 600 + 150 ( 0.54) +95 (0.54 )

2

]

V 1=0.54646

m3 kg

w=∫ Pdv

2 ¿ 600+150 v 1+95 v 1 ( v 1−v 2 )

¿ 32.6701

55. Air is used in a Carnot engine where the volumes at the beginning of the constant temperature her addition and the constant temperature heat rejection are 0.356m3 and 5.573m3 respectively. Calculate the thermal efficiency.

rk =

v 1 5.573 = =15.65 v 2 0.356

e=1−

1 r k k −1

¿ 0.66722

56. An air standard Otto cycle has a compression ratio of 8. At the beginning of compression the pressure and temperature of air are 100kpa and 25C respectively. The heat added is 1500 KJ/kg. Calculate the mean effective pressure in kpa.

p1 v 1=mRT 1 100 v 1=( 0.287 )( 25 + 273 ) ¿ 1131.9359

m3 v 1=0.85526 kg

e=1−

1 r k k −1

eQA

Pm=

¿ 0.5647247

(

V 1 1−

1 rk

)

¿ 1132 kPa

57. The cycle work of an air standard Otto engine is 1000KJ/kg. The maximum cycle temperature is 3200C and the temperature at the end of isentropic compression is 650C Calculate the compression Ratio of the engine.

¿ 0.716 (3200−650 )

Q ∆=mCV ∆ T

0.5477 =1−

¿ 1825.8

KJ kg

e=

1000 w = QA 1825.8

¿ 0.547705

e=1−

1 r k k −1

1 rk

1.4− 1

58. Two Carnot engines -A and B operate in series between a high -temperature reservoir at 1200K a low temperature reservoir at 450K. Engine A rejects heat to engine B, which in turn rejects heat the low temperature reservoir. The heat received by engine A is 600KJ. The two engines have Equal thermal efficiencies. Calculate the work of engine B in KJ.

450 TL ¿ 1− 1200 TH QR ∆=225 KJ =Q ∆ B e=1−

¿ 0.625

eA =eB

eA=QA ∆−

QR ∆ QR ∆ =0.625 =600− 600 QA ∆

59. A heat engine has a thermal efficiency of 45%. How much power does the engine produce when heat is transferred into it at a rate of 10^9 KJ/hr? Q= Q=10

(

KJ hr hr 3600 s

MW )( 1000 kw )

¿ 277.7778 MW

e=

w QA

0.45=

W 277.78

W =125 M

60. The thermodynamic efficiency of a heat engine that rejects heat at a rate of 20MW when heat is supplied to it at a rate of 60MW is:

e=

QA−QR QA

¿

60 −20 60

¿ 0.6667=66.67 %

61. A Carnot engine operates using a 5 C energy reservoir and a 27C energy reservoir. The thermodynamic efficiency of this engine is:

e=

T 1−T 2 527−27 = 527 + 273 T1

¿ 0.625

¿ 62.50 %

62. An inventor claims to have created a heat engine which produces 10 Kw of power for a 15 KW input while operating between energy reservoirs at 27°C and 427°C. Is this claim valid?

e=

w 10 = =66.67 % Q 15

e=

T 1−T 2 427 −27 = 427 + 273 T1

¿ 0.5714=57.14 %

63. An Otto Cycle has a compression ratio of 8 and a maximum temperature of 627°C. At the beginning of the compression stroke, the pressure and the temperature of the working fluid are 100kPa and 27°C. Based upon the cold-air standard analysis assumption, how much work is produced per cycle completion?

rk =8

¿ ( 27 + 273) ( 8 ) 1.4 −1

T 2=T 1 ( rk ) k−1

¿ 689.2190 K =416.2190C

273 1.4 − 1 = (8 ) T 4=391.7478 K =118.75 C T4 ¿ ( 0.716) ( 627 −416.22) −(0.716 )(118.75 −27 ) 627+

64.

w=QA −QR

A diesel cycle has a compression ratio of 8 and cut-off ratio of 2.5. Find the cycle efficiency.

rk =6 rc=2 65.

t3 k−1 =( rk ) t4

e=1−

[

1 1 r ck− k −1 k ( rc−1 ) rk

)

¿ 0.4597

¿ 45.97 %

A diesel cycle, has a compression ratio of 6 and cutoff ratio of 2. If heat added is 1500KJ, find the heat rejected.

rk =6 rc=2

1

e=1−

rk

k −1

[

r ck−

1 k ( rc−1 )

)

¿ 0.4283

e=

QA−QR QA

0.4283=

1500−QR 1500

QR=857.60 KJ

66. A diesel has an initial temperature of 30°C, compression ratio of 14 and cut-off ratio of 5.Find the maximum temperature of the cycle.

rk =14 rc =2.5

1¿ 2

T 3= (870.7507 ) (5 )

1.4 −1

T 2= (30 + 273 ) ( 1.4 )

¿ 870.7507 k =597.75 k

2 ¿3

P=C

T3 V3 = T2 V2

¿ 4353.7536 K=4080.75 C

67. In an air standard diesel cycle, compression starts at 100Kpa and 300°K. The compression ratio is 12 to 1. The maximum cycle temperature is 2000°K. Determine the cycle efficiency.

1.4 − 1

T 2= (300 )(12)

¿ 810.5760 K

1 1 k r c k− e=1− k k ( rc −1) r k −1

[

]

2 ¿3

2000 rc = 810.5760

P=C

¿ 0.5422

¿ 2.4674

¿ 54.22%

68. A diesel engine is operating on a 4-stroke cycle, has a heat rate of 10,000 KJ/KW-hr brake. The compression ratio is 10. The cut -off ratio is 2.Using k=1.32, what is the brake engine efficiency.

rk =10; rc=2 ; k =1.32 ee=

e=1−

0.36 eb = =0.787207 e 0.4573

[

1 1 r cK − k −1 rc−1 ( ) k rk

]

¿ 0.4573

eb =1 kw −hr

3600 =0.36 10000

¿ 78.72%

69. Determine the air -standard efficiency of an engine operating on the diesel cycle with clearance of 6% when the suction pressure is 100 Kpa and the fuel is injected for 5% of the stroke. Assume k=1.4

v 3− v 2=0.05 VD

e=1−

70.

e=

V 2=0.06 VD

[

1 1 k rc − k −1 k ( rc−1 ) rk

]

rc =

v 3 0.11 VD =1.833 = v 2 0.06 VD

¿ 0.636828

rk =

1+c 1+0.06 =17.67 = 0.06 c

¿ 63.68 %

A diesel cycle has a cycle efficiency of 58% if heat added is 1600 KJ/kg, find the work.

w QA

0.58=

W 1600

W =928 KJ

71. A diesel cycle has a compression ratio of 8 and Initial temperature of 34°C If maximum temperature of the cycle Is 2000°K, find the cycle efficiency.

8 V 3 T3 ¿ rc = = rk =8 ¿ 705.30 k=432.30C ¿ V 2 T2 T 2=(34 + 273)¿ 1 1 e=1− k −1 r c k − 44.07 % ¿ 0.440654 k ( rc−1 ) rk

[

¿

2000 705.30

¿ 2.8357

]

72. The compression ratio of an air-standard gasoline engine is 8. The heat added is 1850 KJ/kg. If the initial pressure and temperature are 100Kpa and 25°c respectively. Determine the temperature in °C at the end of the isentropic expansion.

v 3 t 3 2261.62 + 273 = = 684.62 v2 t2 2261.62 + 273 1.4− 1 = (2.1609 ) T4 rc =

¿ 3.7022

3¿4

ℜ=

8 rk = =2.1609 rc 3.7022

T3 k−1 =( ℜ ) T4

T 4=1862.3535 K =1589.35C

73. Calculate the thermal efficiency of an air-standard diesel cycle operating with a compression ratio of 23 and cut off ratio of 2.25

rk =23 ; rc=2.25

e=1−

1 1 k rc − rc −1) ( k r k −1 k

[

]

¿ 0.6557

¿ 65.57 %

74. Calculate the thermal efficiency of an air standard limited pressure cycle with a compression ratio of 23, initial pressure and temperature of 100kPa an 25°c respectively, a pressure limited of 12 MPa, and total heat input of 1500KJ/kg of air.

¿ ( 25 + 273) ( 23)1.4 −1

T 2=t 1( rk )k−1

1¿ 2

¿ 1044.4975 K =771.50 C 2 ¿3 QA=mCP ∆ T 2271.4975 + 273 v3 t3 ¿ T 3=2271.4975C rc = = ¿ 2.4361 1044.4975 v4 t2

1500=( 1 ) (T 3−771.50C )

e=1−

[

1 1 k rc − k −1 k ( rc−1 ) rk

]

¿ 0.6483

75. Calculate the thermal efficiency of an air- standard diesel cycle with minimum temperature and pressure of 20°cs and 100kPa, and maximum temperature of 1000°c. The heat added is 800KJ/kg.

T2 200 + 273 =r k1.4 −1 k−1 = (rk ) 20+ 273 T1 1 1 k e=1− k rc − ¿ 0.2154 k ( rc −1 ) r k −1

800=( 1)( 1000 −t 2 )

Q=mCP ∆ T

t 3 1000 + 273 =2.6913 rc = = t 2 200+273

t 2= 200 C

[

rk =3.31

]

76. A three- process cycle operating with 5.0kgs of air as the working substance has the following process: constant volume (1-2) constant pressure (2-3) and constant temperature (3-1).Given that P1 = 100kPa, Ti = 300°k and V1/V3 = 6, determine the heat added in KJ.

v 1=6 v 3

v 1=v 2=6 v 3

QA = mCV (T 2−T 1 )

∆+ p=c∧T 3=T 1

¿ 5 ( 0.716 ) (1800−300 )

T2 V2 = T3 V3 ¿ 530 KJ

T 2 6V 3 = 800 V 3

T 2=1800 K

77. A sterling engine uses an energy source whose temperature is 727°c and energy sink whose temperature is 27°C. How much heat must be added to the engine to produce 1 unit of work?

e=1− 78.

TL TH

¿ 1−

27 + 273 727 + 273

e=

W QA

0.7=

1 QA

QA =1.42857=1.43 units

A Brayton cycle has a compression ratio of 9. Find the cycle efficiency.

rk =9

rp ¿ ¿ v1 =¿ v2

rp=21.6740

1 1.4

9=rp

e=1−

1 rp

79.

¿ 0.7

k−1 k

¿ 0.584756

¿ 58.48 %

A Brayton cycle has an initial air temperature of 30°C. If pressure ratio is 12, find the compressor work.

( rp )k−1 k KJ ¿ 313.28 kg T 2=T 1

1.4 − 1 1.4

( 30 + 273 ) (12) ❑

¿ 616.2829 K =343.28 C

W =mCP ∆ T

¿ ( 1 )(343.28 −30 )

80. A Brayton cycle has an initial air temperature of 27°C and pressure ratio of 8. If maximum temperature is 1400°C, find the heat added in the combustor.

rp=8

1¿ 2

) t 2=t 1( rp

¿ 1 ( 1400 −270.43 ) 81.

¿ ( 27 + 273) (8

1.4 −1 1.4

1.4 − 1 1.4

)

¿ 543.4342 K =270.43C

QA = mCP (T 3−T 2)

¿ 1129.5058 KJ /kg

The maximum temperature of Brayton cycle is 1600°C and its pressure ratio is 6. Find the turbine work.

t3 =( rp) t4

1.4 − 1 1.4

6 ¿ ¿ 1.4 −1 1.4 1600 + 273 =¿ t4

t 4=1122.5382 K =849.56 C

WT =mCp ( t 3−t 4 )

¿ 1 ( 1600 −849.56 )

¿ 750.4418 82.

The compression ratio of Brayton cycle is 10. Find the cycle efficiency. 1

rk =10

v1 =rp k v2

1

rp=25.1189

10=rp 1.4

83. The temperature of four corners of Brayton cycle are t1= 30C, t2 = 400C, t3 = 1300C and t4 = 100C. Find the cycle efficiency.

e=1−

T 4−T 1 T 3−T 2

¿ 1−

100−30 1300− 400

¿ 0.92222

¿ 92.22 %

84. A Brayton cycle has an initial condition of 28C and maximum temperature of 1450C. If pressure ratio is 10, find the backwork ratio.

85. A gas turbine on an air standard Brayton cycle has air entering into the compressor at atmospheric condition and 23C. The pressure ratio is 10 and the maximum temperature in the cycle is 1000C. Compute the cycle efficiency per kg of air in percent.

rp=10

1

e=1−

rp

k−1 k

¿ 0.4821

¿ 48.21 %

86. Air enter the combustion chamber of a gas turbine unit at 500kpa, 227C and 40m/s. The products of combustion leaves the combustor at 598kpa, 1007C and 150 m/s. Liquid fuel enters with a heating value of 43.000 KJ/kg. The combustor efficiency is 92%. What is the air-fuel ratio? Properties of air. At 500frm h=503.02; At 12864, h 1372.25 KJ/kg

Qfuel=mCP ( T 2−T 1) + ¿ ( 1 )(1007 −227 ) +

V 22 −V 12 2000

1502−402 2000

¿ 790.45

KJ kg air

87. Air enters the compressor of a gas turbine at 102kpa and 320K with a volume flow rate of 6cu. M per sec. The compressor pressure ratio is 10 and its isentropic efficiency is 82%. At the inlet to the turbine, the pressure is 1000kpa and the temperature is 1450K. The turbine has isentropic efficiency of 0.87 and the exit pressure is 102 kPa. On the basis of air standard analysis, what is the thermal efficiency of the cycle in percent? 88. In a gas turbine unit, air enters the combustion chamber at 550 kpa, 227C and 43m/s. The products of combustion leave the combustor at 511 kpa, 1004C and 140m/s. Liquid fuel enters with a heating value of 43,000 KJ/kg. For fuel air ratio of 0.0229, what is the combustor efficiency of the unit in percent?

heat supplied by fuel=mfQh

¿ ( 0.0229) (43000 )

1 2 2 Q=Cp ( T 2−T 1 )+ ( V 2 −V 1 ) 2 ¿ 785.8755

KJ KG

¿ 984.7

1402− 432 1000

KJ air kg

Q=heat absorbed by fuel

)

1 Q = (1 )( 1004 −227 ) + ¿ 2 m 785.9 Combustor Eff = ¿ 0.7981 984.7

89. An ideal gas turbine operates with a pressure of 8 and temperature limits of 20C and 1000C. The energy input in the high temperature heat exchanger is 200kw. Determine the air flow rate in kg/hr....