Thermal Efficiency of an Ideal Simple Brayton (Gas Turbine) Cycle For an ideal gas turbine working on the simple Brayton cycle with perfect intercooling/recuperation absent, the thermal efficiency in terms of pressure ratio r_p and specific heat ratio gamma is eta_th = 1 − 1 / r_p^((gamma − 1)/gamma).

Introduction / Context:
The Brayton (gas-turbine) cycle is characterized by two isentropic and two constant-pressure processes. A widely used closed-form expression relates its ideal thermal efficiency to the compressor pressure ratio and the specific heat ratio gamma, enabling quick preliminary design and performance estimates.

Given Data / Assumptions:

• Simple, ideal Brayton cycle (no regeneration, intercooling, or reheat).
• Isentropic compressor and turbine; isobaric heat addition and rejection.
• Ideal-gas behavior with constant gamma.

Concept / Approach:

Using the isentropic temperature–pressure relation T2/T1 = r_p^{(gamma−1)/gamma} and T3/T4 = r_p^{(gamma−1)/gamma}, one finds that the thermal efficiency depends only on r_p and gamma: eta_th = 1 − 1 / r_p^{(gamma−1)/gamma}. This shows efficiency increases with pressure ratio for the ideal case (up to limits when real effects are included).

Step-by-Step Solution:

Verification / Alternative check:

An equivalent temperature-based form is eta_th = 1 − (T4 − T1)/(T3 − T2); substituting the isentropic equalities collapses it to the pressure-ratio expression above. Both are correct under ideal assumptions.

Why Other Options Are Wrong:

Option (b) omits the isentropic links; (c) is not a definition of thermal efficiency; (d) is correct in temperature form but only after invoking isentropic equalities—since the question asks for the standard r_p–gamma form, (a) is the most explicit; (e) is a mechanical efficiency-like ratio, not thermal efficiency.

Common Pitfalls:

Applying the ideal formula at high turbine inlet temperatures without accounting for variable properties and component efficiencies; confusing cycle thermal efficiency with component isentropic efficiencies.

Final Answer:

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