Answer:
(a) the mass flow rate of air is 5.351 kg/s
(b) the input power required is 2090.786 kW
Explanation:
Given;
initial pressure, P₁ = 100 kPa
initial temperature, T₁ = 20 °C
Final pressure, P₂ = 1.8 MPa
Final temperature, T₂ = 400 °C
Inlet area of the compressor = 0.15 m²
outlet area of compressor = 0.078 m²
velocity of air = 30 m/s
Part (a) mass flow rate of air through the inlet
Mass flow rate = Area x velocity = density x volumetric rate
m = Av = ρV
from ideal gas law, PV = nRT and ρ = m/V
substitute these values in the above equations, we will have;
![m = \frac{PAv}{RT}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7BPAv%7D%7BRT%7D)
![m = \frac{100000*0.15*30}{287*(20+273)}= 5.351 \ kg/s](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B100000%2A0.15%2A30%7D%7B287%2A%2820%2B273%29%7D%3D%205.351%20%5C%20kg%2Fs)
Part (b) the required power input
![W + m(h_1+\frac{v_1{^2}}{2}) = mh_2](https://tex.z-dn.net/?f=W%20%2B%20m%28h_1%2B%5Cfrac%7Bv_1%7B%5E2%7D%7D%7B2%7D%29%20%3D%20mh_2)
where;
W is the input power
m is the mass flow rate
h₁ is the initial enthalpy
h₂ is the final enthalpy
initial and final enthalpy are obtained from steam table using interpolation;
h₁ = 293.166 kJ
h₂ = 684.344 kJ
![W + m(h_1+\frac{v_1{^2}}{2}) = mh_2\\\\W = mh_2 - m(h_1+\frac{v_1{^2}}{2})\\\\W = 5.351 ( 684.344) - 5.351 (293.166 + \frac{30^2}{2000}) \\\\W = 3661.925 \ kW -1571.139 \ kW\\\\W = 2090.786 \ kW](https://tex.z-dn.net/?f=W%20%2B%20m%28h_1%2B%5Cfrac%7Bv_1%7B%5E2%7D%7D%7B2%7D%29%20%3D%20mh_2%5C%5C%5C%5CW%20%3D%20mh_2%20-%20m%28h_1%2B%5Cfrac%7Bv_1%7B%5E2%7D%7D%7B2%7D%29%5C%5C%5C%5CW%20%3D%205.351%20%28%20684.344%29%20-%205.351%20%28293.166%20%2B%20%5Cfrac%7B30%5E2%7D%7B2000%7D%29%20%5C%5C%5C%5CW%20%3D%203661.925%20%5C%20kW%20-1571.139%20%5C%20kW%5C%5C%5C%5CW%20%3D%202090.786%20%5C%20kW)