Consider the adiabatic air compressor
shown below. 















a.) Calculate the minimum power input required and T_{2}





b.) The outlet temperature from a real, adiabatic compressor that accomplishes the same
compression is 520K. Calculate the actual power
input and the isentropic efficiency of the real compressor. 























Read : 
Determine S^{o}(T_{2}) for an isentropic process and then interpolate to obtain both T_{2S} and H_{2S}. Then, an energy balance will give you (W_{S})_{min}. Use the isentropic efficiency and (W_{S})_{min} to determine (W_{S})_{act}. 


















Given: 
m 
9.5 
kg/s 

Find: 
a.) 
(W_{S})_{min} 
??? 
kW 





P_{1} 
110 
kPa 



T_{2S} 
??? 
K 





T_{1} 
310 
K 


b.) 
(W_{S})_{act} 
??? 
kW 





P_{2} 
550 
kPa 



h_{S, comp} 
??? 






T_{2, part (b)} 
520 
K 





^{} 


















Diagram: 
The diagram in the
problem statement is adequate. 














Assumptions: 
1  
The compressor operates at steadystate and there is no significant heat transfer. 






2  
Kinetic and potential
energy changes are negligible. 






3  
Air is modeled as an
ideal gas. 


















Equations
/ Data / Solve: 

























Part a.) 
An isentropic
compressor requires the minimum power input. 



















We can determine the isentropic work by applying the 1st Law to an isentropic compressor that takes in
the same feed and yields an effluent at the same pressure. 



















For a steadystate, singleinlet, single outlet system with negligible heat transfer, kinetic and potential energy changes, the 1st Law is: 
























Eqn 1 















The entropy
change for this process can determined using the 1st Gibbs Equation in terms of the Ideal Gas Entropy Function. 
























Eqn 2 



















We can also apply Eqn 2 to our
hypothetical, isentropic compressor: 

Eqn 3 















We can solve Eqn 3 for the unknown S^{o}_{T2S} : 


Eqn 4 



















We can evaluate S^{o}_{T1} using the Ideal Gas Property Tables: 
S^{o}(T_{1}) 
0.0061681 
kJ/kgK 



















We can get H^{o}_{T1} while we are looking
in the Ideal Gas Property Tables because we will need it later when we evaluate Eqn 1. 











H_{1} 
87.410 
kJ/kg 



















Now, we can plug values into Eqn 4 : 



R 
8.314 
kJ/kmolK 










MW 
28.97 
kg/kmol 











S^{o}(T_{2S}) 
0.46806 
kJ/kgK 



















Now, we can use S^{o}_{T2S} and the Ideal Gas Property Tables to
determine T_{2S} and H_{2S} by interpolation. 



















T (K) 
H (kJ/kg) 
S^{o} (kJ/kgK) 










470 
260.49 
0.46258 











T_{2S} 
H_{2S} 
0.46806 



T_{2S} 
472.50 
K 





480 
270.88 
0.48445 


H_{2S} 
263.09 
kJ/kg 


















Now, we can plug
values back into Eqn 1 : 

(W_{S})_{min} 
1669.0 
kW 


















Part b.) 
We can determine the actual power input
for the compressor by
applying the 1st Law
to the real compressor, just as we did in Eqn 1 for the isentropic compressor. 

























Eqn 5 



















We can evaluate H^{o}_{T2} using the Ideal Gas Property Tables: 
T_{2, part (b)} 
520 
K 











H_{2} 
312.65 
kJ/kg 



















Now, we can evaluate W_{S,act} using Eqn 5: 

(W_{S})_{act} 
2139.78 
kW 



















The isentropic efficiency of a compressor is defined by: 


Eqn 6 



















Since we determined
the isentropic work in part
(a) and the actual work in
part (b), we are ready
to plug numbers into Eqn 6 and wrap up this problem. 

























h_{S, comp} 
78.00% 



















Verify: 
Check the Ideal Gas assumption: 


V_{1}
= 
23.43 
L/mole 










V_{2}
= 
7.86 
L/mole 



















Since air can be considered to be a diatomic gas and both molar volumes are greater than 5 L/mole, it is accurate to treat the air as an ideal
gas. 


















Answers : 
Part a.) 
(W_{S})_{min} 
1670 
kW 

Part b.) 
(W_{S})_{act} 
2140 
kW 






T_{2S} 
473 
K 


h_{S, comp} 
78.0% 
























































































