Triangular (V-Notch) Weir – Sensitivity of Discharge to Head Measurement For a triangular notch with fixed angle and coefficient, the relationship between fractional discharge error and fractional head error is (dQ/Q) = (5/2) * (dH/H).

Introduction:
Discharge over a triangular (V-notch) weir is very sensitive to the head reading. Quantifying this sensitivity helps set instrumentation accuracy and staff-gauge resolution for field measurements.

Given Data / Assumptions:

• Standard sharp-crested V-notch with fixed notch angle and constant Cd.
• Free nappe with proper aeration; negligible approach velocity correction.
• Small measurement errors relative to the head value (differential approximation).

Concept / Approach:

For a V-notch, Q ∝ H^(5/2). Taking logarithms and differentiating gives dQ/Q = (5/2) * (dH/H) when Cd and geometric factors are treated as constants. Thus, a 1% error in head yields about a 2.5% error in discharge.

Step-by-Step Solution:

Verification / Alternative check:

Repeating the same steps for rectangular weirs (Q ∝ H^(3/2)) yields dQ/Q = (3/2) * dH/H, illustrating the stronger sensitivity of triangular notches.

Why Other Options Are Wrong:

3/2 corresponds to rectangular weirs. 1/2 and 2 underestimate the exponent effect. 5/3 is not derived from the H^(n) dependence of triangular notches.

Common Pitfalls:

Ignoring uncertainty in Cd or notch angle; applying this relation when approach velocity correction is significant, which slightly alters K but not the exponent.

Final Answer:

(dQ/Q) = (5/2) * (dH/H)