1 kg of air undergoes a polytropic compression from 1 bar and 290 K to 6 bar and 400 K.

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1 kg of air undergoes a polytropic compression from 1 bar and 290 K to 6 bar and 400 K. If the temperature and pressure of the surroundings are 290 K and 1 bar respectively, determine :

(i) The irreversibility ;

(ii) The effectiveness.

Take for air : cp = 1.005 kJ/kg K, cv = 0.718 kJ/kg K, R = 0.287 kJ/kg K

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Mass of air, m = 1 kg

Initial temperature, T1 = T0 = 290 K

Final temperature, T2 = 400 K

Initial pressure, p1 = p0 = 1 bar

Final pressure, p2 = 6 bar.

(i) The irreversibility, I :

We know that reversible work,

Wrev. = Change in internal energy – T0 × Change in entropy

= 134.37 kJ/kg (–) [–ve sign indicates that work is done on the air] Actual work is given by

The index n for the compression is given by :

(–ve sign means that the work is done on the air)

Now, irreversibility = Wrev – Wactual

= – 134.37 – (– 144.8) = 10.43 kJ.

(ii) The effectiveness, ∈ :

∈ = $\cfrac{W_{rev}}{W_{actual}}$$\cfrac{-134.37}{-144.8}$ = 0.928 or 92.8%.