Ericsson cycle

Example

The ideal Ericsson cycle is a thermodynamic cycle of $n$ moles of an ideal gas. It runs between a hot reservoir at temperature $T_h$ and a cold reservoir at temperature $T_c < T_h$. Let $p_L < p_H$ be two pressures.

1. Isothermal compression at $T_c$ from $p_L$ to $p_H$.
2. Isobaric heating at $p_H$ from $T_c$ to $T_h$.
3. Isothermal expansion at $T_h$ from $p_H$ to $p_L$.
4. Isobaric cooling at $p_L$ from $T_h$ to $T_c$.

The gas returns to its initial state, so this is a cycle. For an ideal gas the internal energy depends only on temperature, so on each isotherm the heat equals the work. The gas takes in $Q_h = nRT_h \ln(p_H/p_L)$ on the hot isotherm and gives off $Q_c = nRT_c \ln(p_H/p_L)$ on the cold isotherm. The two isobaric steps exchange the same heat $nC_p(T_h - T_c)$ in opposite directions, where $C_p$ is the molar heat capacity at constant pressure. With an ideal regenerator this heat stays inside the engine, and the thermal efficiency is $$\eta = \frac{Q_h - Q_c}{Q_h} = 1 - \frac{T_c}{T_h}.$$