Brayton cycle

Example

Take $n$ moles of an ideal gas with constant molar heat capacity $c_p$ at constant pressure and constant heat capacity ratio $\gamma > 1$. Let $p_2 > p_1$ be two pressures, with pressure ratio $r_p = p_2/p_1$. The ideal Brayton cycle has four steps, with temperatures $T_1, T_2, T_3, T_4$ at their start points:

1. isentropic compression from $p_1$ to $p_2$;
2. isobaric heat addition at $p_2$, absorbing $Q_H = n c_p (T_3 - T_2)$;
3. isentropic expansion from $p_2$ to $p_1$;
4. isobaric heat rejection at $p_1$, rejecting $Q_C = n c_p (T_4 - T_1)$, back to the initial state.

The gas returns to its initial state, so this is a thermodynamic cycle. On an isentrope of an ideal gas $T p^{(1-\gamma)/\gamma}$ is constant, so $T_2/T_1 = T_3/T_4 = r_p^{(\gamma-1)/\gamma}$. Hence $T_3 - T_2 = r_p^{(\gamma-1)/\gamma} (T_4 - T_1)$, and the thermal efficiency is $$\eta = 1 - \frac{Q_C}{Q_H} = 1 - r_p^{-(\gamma - 1)/\gamma}.$$ For $r_p = 10$ and $\gamma = 1.4$, $\eta = 1 - 10^{-2/7} \approx 0.48$.