Ideal Brayton (gas turbine) cycle — overall thermal efficiency vs. pressure ratio For an ideal gas turbine plant operating on the Brayton cycle with isentropic compressor and turbine, the overall thermal efficiency (eta_th) can be expressed purely as a function of the compressor pressure ratio r and specific heat ratio gamma. Which expression is correct?

Introduction / Context:The Brayton (Joule) cycle models the ideal gas turbine. Its thermal efficiency relates closely to the compressor pressure ratio r and the specific heat ratio gamma for an ideal gas with isentropic compression/expansion and constant-pressure heat addition and rejection. Understanding this dependence helps with preliminary cycle sizing and estimating benefits of increasing pressure ratio.

Given Data / Assumptions:

• Ideal gas with constant gamma.
• Isentropic compressor and turbine.
• Heat addition and rejection at constant pressure.
• Negligible pressure losses in combustor, ducts, and nozzle.

Concept / Approach:For an ideal Brayton cycle, temperature ratios across isentropic devices depend only on pressure ratio and gamma. The compressor exit to inlet temperature ratio is T2/T1 = r^((gamma - 1)/gamma) and the turbine exit to inlet temperature ratio is its reciprocal for the same r. The thermal efficiency simplifies to eta_th = 1 - (T4 - T1)/(T3 - T2). With the isentropic relations and equal pressure ratios, this reduces to a function of r and gamma alone.

Step-by-Step Solution:Use isentropic relation: T2/T1 = r^((gamma - 1)/gamma).For the matched turbine: T3/T4 = r^((gamma - 1)/gamma) → T4/T3 = 1 / r^((gamma - 1)/gamma).Ideal Brayton efficiency simplifies to eta_th = 1 - (T1/T2) = 1 - 1 / r^((gamma - 1)/gamma).Therefore the correct expression is eta_th = 1 - 1 / r^((gamma - 1)/gamma).

Verification / Alternative check:As r increases, r^((gamma - 1)/gamma) increases, making 1 / r^((gamma - 1)/gamma) smaller; hence efficiency rises, which matches known Brayton behavior (up to metal temperature limits).

Why Other Options Are Wrong:(b) and (d) do not include gamma and are dimensionally and physically inconsistent. (c) would predict negative efficiency for r > 1. (e) represents the loss term alone, not 1 minus that term.

Common Pitfalls:Confusing r^((gamma - 1)/gamma) with r^(gamma - 1), or forgetting the reciprocal in the final expression.

Final Answer:eta_th = 1 - 1 / r^((gamma - 1)/gamma)