Question 4.19: Problem: Oxygen enters a compressor at a pressure of 100 kPa...
Problem: Oxygen enters a compressor at a pressure of 100 kPa, temperature of 300 K and a velocity of 5 m / s with a mass flow rate of 0.5 kg / s. The compressor is cooled at a rate of 1 kW. The gas exits at 500 kPa and 400 K with a velocity of 3 m / s. Find the work input to the compressor.
Find: Shaft work input to the compressor \dot{W} .
Known: Inlet pressure P_1 = 100 kPa, inlet temperature T_1 = 300 K, inlet velocity \pmb{V}_1 = 5 m / s, outlet pressure P_2 = 500 kPa, outlet temperature T_2 = 400 K, outlet velocity \pmb{V}_2 = 3 m / s, heat loss \dot{Q} = −1 kW.
Assumptions: Oxygen is an ideal gas with constant specific heat, change in potential energy is negligible, kinetic energy changes and heat losses are not negligible.
Governing Equation:
Energy rate balance \dot{Q} + \dot{W} =\dot{m} \left[\left(h_2-h_1\right) + \frac{\pmb{V}_2^2-\pmb{V}_1^2}{2} + g(z_2-z_1) \right]
Properties: Average temperature of oxygen is T_{avg} = ( T_2 + T_1 ) / 2 = ( 300 \ K + 400 \ K ) / 2 = 350 \ K ; at 350 K specific heat of oxygen is c_p = 0.928 kJ / kgK (Appendix 4).
\dot{Q} + \dot{W} =\dot{m} \left[c_p\left(T_2-T_1\right) + \frac{\pmb{V}_2^2-\pmb{V}_1^2}{2} + \underbrace{g(z_2-z_1)}_{=0} \right]
-1 \ kW + \dot{W} =0.5 \ kg/s\left[0.928 \ kJ/kgK \left(400 \ K – 300 \ K\right)+\frac{\left(5 \ m/s\right)^2-\left(3 \ m/s\right)^2 }{2} \times \frac{1}{1000 \ J/kJ} \right]
\dot{W} =1 \ kW + 0.5 \ kg / s \times \left[92.8 \ kJ/kg + 0.008 \ kJ/kg\right] =47.404 \ kW
Answer: The compressor requires 47.4 kW to drive it. Note that the contribution due to the change in kinetic energy was very small. Neglecting it would have made little difference to the final answer.