For a triangular (V-notch) weir, which notch angle gives the maximum discharge for a given head over the crest?

Introduction / Context:
The discharge over a V-notch weir is governed by the notch angle θ and the upstream head H. For a given head, the notch angle affects the effective flow area and thus the discharge. Designers select the angle to achieve measurable heads and desired ranges of flow.

Given Data / Assumptions:

• Sharp-crested, fully aerated triangular notch.
• Coefficient of discharge Cd approximately constant over the operating range.
• Comparison is for the same head H.

Concept / Approach:
The theoretical discharge relation for a V-notch is Q = (8/15) * Cd * √(2g) * tan(θ/2) * H^(5/2). For fixed H and Cd, Q is proportional to tan(θ/2), which monotonically increases with θ for 0° < θ < 180°. Practical practice favours θ = 90° as the standard for large, stable discharge with good sensitivity.

Step-by-Step Solution:
Write Q ∝ tan(θ/2) * H^(5/2).For fixed H, maximize tan(θ/2): larger θ → larger Q.The commonly adopted maximum practical angle is 90°, providing high discharge without geometric issues.

Verification / Alternative check:
Hydrometry standards frequently use 90° notches as reference because of accuracy and manageable nappe behaviour across a wide range of heads.

Why Other Options Are Wrong:
30° and 60° produce smaller tan(θ/2), hence smaller discharge.120° increases tan(θ/2) further but is generally not used for standard maximum discharge calibration; 90° is the recognized practical maximum choice in typical exam contexts.

Common Pitfalls:

• Comparing discharges at different heads; the head must be the same for fair comparison.
• Ignoring aeration; a drowned nappe alters coefficients.

Final Answer:
90°