6.2: Irreversible Cycles
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Up to this point, we have discussed reversible cycles only. Notice that the heat that enters the definition of entropy (Definition: Entropy) is the heat exchanged at reversible conditions since it is only at those conditions that the right-hand side of Equation 6.1.5 becomes a state function. What happens when we face an irreversible cycle? The efficiency of a Carnot cycle in Equation 5.3.3 is the maximum efficiency that an idealized thermodynamic cycle can reach. As such, any irreversible cycle will incontrovertibly have an efficiency smaller than the maximum efficiency of the idealized Carnot cycle. Therefore, Equation 6.1.1 for an irreversible cycle will not hold anymore and must be rewritten as:
\[ \overbrace{1+\dfrac{Q_3}{Q_1}}^{\varepsilon_{\mathrm{IRR}}} < \overbrace{1-\dfrac{T_l}{T_h}}^{\varepsilon_{\mathrm{REV}}}, \label{6.2.1} \]
and, following the same procedure used in section 6.1, we can rewrite Equation \ref{6.2.1} as:
\[ \dfrac{Q^{\text{IRR}}_3}{Q^{\text{IRR}}_1} < - \dfrac{T_l}{T_h} \longrightarrow \dfrac{Q^{\text{IRR}}_3}{T_l} + \dfrac{Q^{\text{IRR}}_1}{T_h} < 0 \longrightarrow \sum_i \dfrac{Q_{\text{IRR}}}{T_i} < 0, \nonumber \]
which can be generalized using calculus to:
\[ \oint \dfrac{đQ_{\mathrm{IRR}}}{T} < 0. \label{6.2.3} \]
Putting eqs. 6.1.6 and \ref{6.2.3} together, we obtain:
\[ \oint \dfrac{đQ}{T} \leq 0, \label{6.2.4} \]
where the equal sign holds for reversible transformations exclusively, while the inequality sign holds for irreversible ones. Equation \ref{6.2.4} is known as Clausius inequality.