Air is compressed from 100 kPa and 150C to a pressure of 1000 kPa while being cooled at a rate of 20 kJ/kg by circulating water through the compressor casing. The volume flow rate of the air at the inlet conditions is 140 m3/min, and the power input to the compressor is 520 kW Determine: (a) The mass flow rate of the air, and (b) The temperature at the compressor exit.

a) 115.7 kg/min

b) 240 C

Explanation:

To determine the mass flow rate we need the specific volume of the air. We can use the gas state equation for that:

p * v = R (air) * T (no mass term because we want specific volume)

v = R * T / p

The specific R for air is R = 287 J / (kg * K)

Also 150 C = 423 K

Then

v = 287 * 423 / 100e3 = 1.21 m^3/kg

Them the mass flow is

G / v = 140 / 1.21 = 115.7 kg/min

The flow rate can also be expressed as 115.7 / 60 = 1.93 kg/s

Then we find that the compressor applies a work of

L = 520 / 1.93 = 269.4 kJ/kg to the air

By enthalpy:

H2 = H1 + Q + L

H2 = cv * T2 + p2 * v2

H1 = cv * T1 + p1 * v1

Then

cv * T2 + p2 * v2 = cv * T1 + p1 * v1 + Q + L

Also

(p1 * v1) / T1 = (p2 * v2) / T2

p2 * v2 = (p1 * v1 * T2) / T1

Therefore:

cv * T2 + (p1 * v1 * T2) / T1 = cv * T1 + p1 * v1 + Q + L

T2 * (cv + (p1 * v1) / T1 = cv * T1 + p1 * v1 + Q + L

T2 = (cv * T1 + p1 * v1 + Q + L) / ((cv + (p1 * v1) / T1)

The cv of air is cv = 0.72 kJ / (kg*K)

T2 = (0.72 * 423 + 100e3 * 1.21 - 20 + 269.4) / ((0.72 + (100e3 * 1.21) / 423) = 513 K = 240 C