Air is contained in a frictionless cylinder/piston system and is compressed from

Question

Air is contained in a frictionless cylinder/piston system and is compressed from 2 bar and 25C to a new pressure of 7 bar according to P*V^1.5=constant. If the mass of air is 5kg, What is the work done of the air and the heat supplied during the compression? Additional data are as follows: air molecular weight is 29 and Cp=1.005 kj/kg

Air is contained in a frictionless cylinder/piston system and is compressed from 2 bar and 25C to a new pressure of 7 bar according to P*V^1.5=constant. If the mass of air is 5kg, What is the work done of the air and the heat supplied during the compression? Additional data are as follows: air molecular weight is 29 and Cp=1.005 kj/kg

Explanation / Answer

Note that (MM)PV = mRT --> V = mRT/[(MM)P]          
          
Thus, when T = 25 C = 298 K, P = 2 bar,          
          
V (2 bar)=    0.002135966   m^3  
          
Thus,          
          
PV^1.5 =    19.743385      
          
Thus, at P = 7 bar,          
          
V(7 bar) =    0.000926579   m^3  
          
Thus, as W = Integral[PdV], and P = 19.74338V^-1.5, then          
          
W = Integral[19.74338V^-1.5 dV] = 39.49V^(-0.5)|(0.002125 m^2, 0.000927 m^3)          
          
W =   -443    J   [ANSWER, WORK DONE BY AIR]
          
Now, solving for the new T after the compression, T = [(MM)PV/(mR)]          
          
Thus,          
          
T =    452.4517568   K  
          

Thus, delta(U) = m * cp          
          
delta(U) =    5025   J  
          
Thus, as Q = delta(U) + W,          
          
Q =    4580    J   [ANSWER, HEAT SUPPLIED]

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