Show that the heat rejected per stage per kg of air in a reciprocating compressor with perfect intercooling is given by ((Cp+Cv(y-n÷n-1))(T2-T1)

Total heat rejected by compressor = Heat rejected by compression + Heat rejected by intercooler

Heat rejected by compression :

Q = (y - n)/(y - 1) • W

Q = (y - n)/(y - 1) • R(T2 - T1)/(n - 1) ... (i)

Heat rejected by intercooler :

Q = Cp(T2 - T1) ... (ii)

Sum of (i) and (ii) :

Q = (i) + (ii)

Q = (y - n)/(y - 1) • R(T2 - T1)/(n - 1) + Cp(T2 - T1) ... (iii)

Remember that :

R = Cp - Cv

y = Cp/Cv

So :

R = yCv - Cv

R = Cv(y - 1) ... (iv)

Now, subtitute (iv) to (iii) :

Q = (y - n)/(y - 1) • Cv(y - 1)(T2 - T1)/(n - 1) + Cp(T2 - T1)

Q = (y - n)/(n - 1) • Cv(T2 - T1) + Cp(T2 - T1)

Q = [Cp + Cv • (y - n)/(n - 1)] • (T2 - T1)

Hope this help