Flow measurement over a triangular (V-notch) weir: If the head over the notch has a 1% measurement error, what is the approximate percentage error in the computed discharge?

Introduction / Context:
Triangular (V-notch) weirs are widely used to measure low flows because they are sensitive to small heads. The discharge–head relationship is nonlinear, so small head errors amplify in discharge calculations. Understanding error propagation helps choose proper gauges and establish calibration procedures.

Given Data / Assumptions:

• Triangular V-notch with standard calibration.
• Head measurement error ΔH/H = 1%.
• Discharge formula follows Q ∝ H^(5/2).

Concept / Approach:

For a V-notch, Q = K * H^(5/2). For small errors, percentage error in Q is obtained by logarithmic differentiation: dQ/Q = (5/2) * dH/H. Thus, a 1% head error produces a 2.5% discharge error. This shows the importance of precise head measurement for V-notches compared to rectangular notches with different exponents.

Step-by-Step Solution:

Verification / Alternative check:

Numerical check: Increase H by 1%; raise to power 2.5 and compare the ratio; the result is ≈ 1.025 for small errors, confirming 2.5%.

Why Other Options Are Wrong:

• 1.0, 1.5, 2.0: Underestimate error amplification from the 2.5 exponent.
• 0.5: Not consistent with the exponent.

Common Pitfalls:

• Using linear error scaling (1:1) instead of exponent-based propagation.
• Neglecting notch coefficient variability, which is a separate systematic effect.

Final Answer:

2.5