Brayton Cycle Efficiency (Pressure Ratio)

The gas turbine's air-standard efficiency depends only on pressure ratio, in the same way the Otto cycle depends only on compression ratio. At $r_p = 8$ with $\gamma = 1.4$ the answer is $1 - 8^{-2/7} = 44.8\%$. Modern industrial machines run pressure ratios of 15 to 25 and aero engines go past 40, and the equation duly promises 55 to 65%.

The catch that the equation cannot express is that efficiency is not the only thing that matters. As pressure ratio climbs, the specific work, meaning the output per kilogram of air, rises to a peak and then falls, because the compressor is eating an ever larger share of what the turbine makes. A machine optimised purely for efficiency needs to swallow far more air for the same output, which means a physically larger and more expensive engine. Real designs sit between the pressure ratio that maximises efficiency and the one that maximises specific work, and the choice depends on whether the customer pays more for fuel or for hardware.

Turbine inlet temperature does not appear in this equation at all, which is the biggest thing the air-standard model hides. In practice it is the dominant design variable: raising it lifts both efficiency and specific work, and the entire history of gas turbine development is metallurgy, film cooling and thermal barrier coatings chasing it upward. It is also why the exhaust is still 500 to 600 °C and why combined-cycle plants bolt a Rankine steam cycle on the back, reaching over 60% overall by making the gas turbine's waste heat someone else's fuel.