Pressure and power for adiabatic compressor

Question

a) Air flowing at 1 m³/s enters an adiabatic compressor at 20°C and 1 bar. It exits at 200°C. The isentropic efficiency of the compressor is 80%. Calculate the exit pressure and the power required.

b) Steam enters a turbine at 10 MPa and 500°C and leaves at 100 kPa. The isentropic efficiency of the turbine is 85%. Calculate the exit temperature and the work generated per kg of steam flowing through.

Answer 1

(a) The isentropic efficiency = (T_exit,s – T_in)/(T_exit,a – T_in) ... subscript s refers to isentropic and a refers to actual temperatures.

0.8 = (T_exit,s – 20)/(200 – 20)

T_{exit, s} = 164 ^oC

Hence, P_exit = (T_in/T_exit,s)^γ/(1-γ)P_in = (283/437)^1/4/(1-1.4) (1 bar) = 4.58 bar

To find the power required,

energy or enthalpy actually changed = C_p (T₂-T₁) = 1(200-20) = 180 kJ/kg

mass flow rate of air = density x volumetric flow rate.

= 0.98 x 1 = 0.98 kg/s

Power required = mass flow rate x enthapy change per unit mass
                       = 0.98 x 180 = 176.4 kW

(b) T_exit,s = (P_exit/P_in)^(γ-1)/γT_in = (100/10000)(1.4-1)/1.4(500)  = 134.13 ^oC

Isentropic efficiency for turbine = (T_exit – T_in)/(T_exit,s – T_in)
                                              = 0.85

T_exit = 0.85(134.12 – 500) + 500 = 189 ^oC

Work generated = -Cp(T_exit – T_in) = 2.28(500 – 189) =   709.8 kJ/kg

Answers, exit temperature is 189 ^oC and work generated is 709 kJ per kg of steam.