When ever a closed system undergoes a cyclic process, the cyclic integral ∮ dQ/T is less than zero (i.e., negative) for an irreversible cyclic process and equal to zero for a reversible cyclic process. The efficiency of a reversible H.E. operating within the temperature T1 & T2 is given by:
η = (Q1– Q2)/Q1 = (T1– T2)/T1 = 1 – (T2/T1)
i.e., 1 – Q2/Q1 = 1 – T2/T1 ; or
Q2/Q1 = T2/T1 ; or; Q1/T1 = Q2/T
Or; Q1/T1 – (– Q2/T2) = 0; Since Q2 is heat rejected so –ive
Q1/T1 + Q2/T2 = 0;
or ∮ dQ/T = 0 for a reversible engine .......(i)
Now the efficiency of an irreversible H.E. operating within the same temperature limit T1 & T2 is given by
= (Q1– Q2)/Q1 < (T1– T2)/T1
i.e., 1 – Q2/Q1 < 1 – T2/T1
or; -Q2/Q1 < T2/T1 ;
or; Q1/T1 < Q2/T2
Or; Q1/T1 – (– Q2/T2) < 0;
Since Q2 is heat rejected so –ive
Q1/T1 + Q2/T2 < 0;
or ∮ dQ/T < 0 for an irreversible engine. ...(ii)
Combine equation (i) and (ii); we get
∮ dQ/T d ≤ 0
The equation for irreversible cyclic process may be written as:
∮ dQ/T + I = 0
I = Amount of irreversibility of a cyclic process
