Ideal Brayton (Gas Turbine) Efficiency in Terms of Pressure Ratio For an ideal simple gas turbine cycle with isentropic compressor and turbine and no regeneration, the thermal efficiency in terms of pressure ratio r and specific heat ratio gamma is eta_th = 1 − 1 / r^((gamma − 1)/gamma).

Introduction / Context:
Expressing Brayton-cycle efficiency in terms of pressure ratio r and gamma provides a rapid sizing and performance estimate for gas turbines in preliminary design. It is derived from isentropic relations and the definition of thermal efficiency as 1 − Q_out/Q_in.

Given Data / Assumptions:

• Ideal gas with constant specific heats.
• Two isentropic and two isobaric processes; no regeneration or intercooling.
• Compressor pressure ratio r = p2/p1 = p3/p4.

Concept / Approach:

Isentropic relations give T2/T1 = r^((gamma−1)/gamma) and T3/T4 = r^((gamma−1)/gamma). Writing Q_in = mCp(T3 − T2) and Q_out = mCp(T4 − T1) and substituting the temperature ratios yields the compact formula eta_th = 1 − 1 / r^((gamma − 1)/gamma).

Step-by-Step Solution:

Verification / Alternative check:

Temperature-form efficiency eta_th = 1 − (T4 − T1)/(T3 − T2) reduces to the r–gamma form under isentropic assumptions, confirming consistency.

Why Other Options Are Wrong:

r^(gamma − 1) and 1 − r^(gamma − 1) lack correct dimensions/limits. r/(gamma − 1) is unrelated. 1 − T1/T3 is Carnot-like and not Brayton’s general result.

Common Pitfalls:

Applying this ideal expression to real engines without accounting for component efficiencies and pressure drops; mixing absolute and gauge pressures when computing r.

Final Answer:

eta_th = 1 − 1 / r^((gamma − 1)/gamma)