Rectangular notch — relationship between relative error in discharge and relative error in head measurement For a sharp-crested rectangular notch where Q ∝ H^(3/2), what is the ratio dQ/Q in terms of dH/H (assuming coefficient and width are constant)?

Introduction / Context:
Error propagation is crucial in open-channel flow measurements using sharp-crested rectangular notches (weirs). Because discharge Q depends on head H with a power law, a small error in head reading amplifies in Q. This question asks for the sensitivity dQ/Q due to dH/H.

Given Data / Assumptions:

• Rectangular sharp-crested notch operating under free, fully aerated flow.
• Discharge relation: Q = K * H^(3/2), where K = (2/3) * C_d * b * sqrt(2g).
• Width b, coefficient C_d, and g treated as constants for small perturbations.
• Differentials are small (linearization valid).

Concept / Approach:
Use logarithmic differentiation of the discharge–head power law. When Q = K * H^n, the relative error relation is dQ/Q = n * dH/H, provided K is constant. Here, n = 3/2 for a rectangular notch.

Step-by-Step Solution:

Verification / Alternative check:
If the head H has a 2% reading uncertainty, the discharge uncertainty from head alone is approximately 3% (because 1.5 * 2% = 3%). This matches field experience where precise head measurement dominates uncertainty.

Why Other Options Are Wrong:

• (1/2) or (2/3): underestimate sensitivity; would correspond to exponents 0.5 or 0.667, not 1.5.
• 3 or 2.5: overestimate sensitivity; these do not match the exponent in the standard formula.

Common Pitfalls:
Forgetting that C_d may vary slightly with H (submergence, aeration), but for first-order estimation we keep it constant. Also, ensure the nappe is properly ventilated to maintain the exponent.

Final Answer:
dQ/Q = (3/2) * (dH/H)