Brayton cycle; Atkinson and Stirling cycles

Brayton cycle efficiency

The Brayton (Joule) cycle is the ideal cycle of a gas turbine. It has isentropic compression 1-2 in a compressor, constant-pressure heat addition 2-3 in a combustor, isentropic expansion 3-4 in a turbine and constant-pressure heat rejection 4-1.

 Air --> [Compressor] --> [Combustor] --> [Turbine] --> Exhaust
             1-2             2-3             3-4        4-1
              \_______________shaft__________/ --> load

$$\begin{aligned} Q_{in} &= c_p (T_3 - T_2), \qquad Q_{out} = c_p (T_4 - T_1) \\ \eta &= 1 - \frac{T_4 - T_1}{T_3 - T_2} \end{aligned}$$

Since $p_2 = p_3$ and $p_1 = p_4$, $T_2/T_1 = T_3/T_4 = r_p^{(\gamma-1)/\gamma}$. Hence $T_4/T_1 = T_3/T_2$ and

$$\eta_{Brayton} = 1 - \frac{T_1}{T_2} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}$$

For $r_p = 8$, $\gamma = 1.4$:

$$\eta = 1 - 8^{-0.2857} = 1 - 0.552 = 0.448$$

Answer: $\eta = 44.8\%$. It depends only on pressure ratio, not on the maximum temperature.

Atkinson cycle

• Same as the Otto cycle (isentropic compression, constant-volume heat addition) but the expansion goes on until the gas reaches atmospheric pressure. The expansion ratio is larger than the compression ratio.
• Heat is rejected at constant pressure.
• It extracts more work from the same fuel, so the efficiency is higher than Otto's.
• It is achieved with a long expansion stroke (original linkage design) or, in modern engines, by closing the inlet valve late (Miller/Atkinson-type timing). The cost is a lower power density, so it is used in hybrid cars.

Stirling cycle

• A closed cycle with an external heat source and a regenerator. Processes: isothermal compression, constant-volume heating (regenerator), isothermal expansion, constant-volume cooling (regenerator).
• With a perfect regenerator its efficiency equals the Carnot efficiency, $1 - T_L/T_H$.
• Heat is supplied from outside, so any fuel or solar heat can be used and combustion is clean and quiet.
• Drawbacks: large size, slow response and sealing problems, so it is rare in vehicles.