Question

Helium gas is compressed steadily from 90 kPa and 25°C to 800 kPa at a rate of 2 kg/min by an adiabatic compressor. The compressor consumes 76 kW of power while operating. What is the isentropic efficiency of this compressor?

Answer 1

Answer:

η = 0.5686

Explanation:

Isentropic compressor efficiency (η):

* η = Isentropic compressor work / real compressor work = Ws / Wreal

∴ Wreal = 76 KW = 76 KJ/s

first law:

• ΔU = Q + W

∴ Q = 0 ....adiabatic compressor

∴ ΔU = CvΔT

∴ W = - PδV

⇒ ΔU = W ,,,,,isentropic process adiabatic

⇒ CvΔT = - PδV

ideal gas:

⇒ (R + Cv / R) Ln(T2/T1) = Ln (P2/P1)

∴ Cv = 3/2 R....monoatomic ideal gas

∴ R = 8.314 E-3 KJ/K.mol

∴ P1 = 90 KPa

∴ P2 = 800 KPa

∴ T1 = 25 °C ≅ 298 K

∴ r He = 2 Kg/min

⇒ (5/2) Ln(T2/T1) = Ln(800/90) = 2.185

⇒ Ln (T2/T1) = 0.874

⇒ T2/T1 = e∧0.874 = 2.3963

⇒ T2 = (2.3963)(298 K) = 714.094 K

⇒ Ws = CvΔT = (3/2 R)(714.094 - 298 ) = 5.1891 KJ/mol

∴ Mw He = 4.0026 g/mol

⇒ Ws = ( 5.1891 KJ/mol)(mol / 4.0026 g)(1000 g/Kg)(2Kg/min)(min/60s)

⇒ Ws = 43.2144 KJ/s

⇒ η = 43.2144 KJ/s / 76 KJ/s = 0.5686