Maximum rotor (diagram) efficiency condition for an impulse stage: The impulse turbine rotor efficiency reaches a maximum value of 0.5 * cos^2(α), where α is the nozzle exit angle, under which blade condition?

Introduction / Context:Rotor (diagram) efficiency in impulse turbines depends on velocity triangles, blade speed ratio, and blade geometry. A key textbook result is the maximum efficiency achievable for given nozzle exit angle α when blades are designed to minimize losses and have symmetric (equiangular) inlet and outlet relative flow angles.

Given Data / Assumptions:

• Impulse staging: pressure drop in nozzles, ideally constant pressure in moving blades.
• Nozzle exit absolute velocity makes an angle α to the wheel tangent.
• Blade friction neglected when velocity coefficient ≈ 1.
• Equiangular blades: inlet and outlet relative flow angles equal in magnitude.

Concept / Approach:Analyzing the velocity triangles and optimizing with respect to blade speed ratio yields a closed-form expression for the maximum rotor (diagram) efficiency. For equiangular and frictionless blades, the optimum gives η_r,max = 0.5 * cos^2 α for the stated condition. This relation highlights the importance of keeping α modest (to increase cos α) and ensuring high-quality, low-loss blade passages to approach the theoretical limit.

Step-by-Step Solution:

Verification / Alternative check:Standard turbine textbooks present this derivation; numerical checks with typical α (for example 20–30 degrees) give plausible maximum efficiencies in the 0.35–0.44 range, consistent with single-stage impulse expectations when only rotor efficiency is considered (overall stage efficiency will be lower after accounting for nozzle and mechanical losses).

Why Other Options Are Wrong:

• Equiangular only: Without assuming frictionless flow (velocity coefficient unity), losses reduce the maximum below the ideal expression.
• Velocity coefficient unity only: Symmetry (equiangular) is also required to reach the stated form.
• Blade solidity 0.65: Solidity influences losses and flow turning but does not by itself produce the stated closed-form maximum.

Common Pitfalls:Confusing rotor (diagram) efficiency with overall stage efficiency; mixing the result η_r,max = cos^2 α (a different assumption set) with the present 0.5 * cos^2 α expression.

Final Answer:blades are equiangular and frictionless