De Laval (impulse) turbine – maximum diagram efficiency For a single-stage De Laval impulse turbine with nozzle angle α (at rotor inlet), the maximum possible diagram (blading) efficiency is:

Introduction / Context:In a simple impulse stage (De Laval turbine), the fixed nozzles convert all pressure drop into a high-velocity jet. The rotor extracts work by turning this jet. For given inlet flow angle α (from the tangent), there exists an optimum blade-speed ratio that maximizes the diagram (blading) efficiency.

Given Data / Assumptions:

• Impulse stage (no pressure drop in rotor).
• Symmetrical or suitably chosen blade angles; negligible losses for the ideal result.
• Velocity coefficient taken as unity for the theoretical maximum.

Concept / Approach:The diagram efficiency η_d is the ratio of work obtained on the blades to the kinetic energy supplied at rotor inlet. Analysis of velocity triangles yields an expression for η_d as a function of the blade-speed ratio and α. Optimizing with respect to blade speed gives the well-known ideal limit η_d,max = cos^2 α.

Step-by-Step Solution:Start from inlet velocity triangle: absolute jet makes angle α with wheel tangent.Express work per kg as product of blade speed and change in whirl component of velocity.Form η_d = (blade work per kg) / (V_inlet^2 / 2).Differentiate with respect to blade-speed ratio and set derivative to zero → optimum condition leads to η_d,max = cos^2 α.

Verification / Alternative check:For α = 20°, cos^2 α ≈ 0.883; for α = 30°, cos^2 α = 0.75, matching classic charts for impulse stage efficiency ceilings.

Why Other Options Are Wrong:

• sin^2 α, tan^2 α, cot^2 α, sec^2 α do not arise from the optimization; they misrepresent the trigonometric dependence.

Common Pitfalls:Confusing α (nozzle angle) with blade inlet/exit angles; the maximum efficiency result is sensitive to which angle is used.

Final Answer:cos^2 α