Rectangular Weir with End Contractions – Francis Discharge Formula For a sharp-crested rectangular weir having n end contractions, the practical Francis form in SI units is Q = 1.84 * (L − 0.1 * n * H) * H^(3/2), where L is crest length and H is head over crest.

Introduction:
The Francis formula is a widely used empirical relation for discharge over sharp-crested rectangular weirs. It corrects the crest length for end contractions and consolidates constants for SI usage.

Given Data / Assumptions:

• Rectangular, sharp-crested weir with free nappe and adequate aeration.
• Head H measured sufficiently upstream to avoid velocity-of-approach error.
• n end contractions (typically 2 for a contracted weir).

Concept / Approach:

The theoretical discharge per unit width from energy considerations scales with H^(3/2). Accounting for contraction and empirical coefficient Cd leads to a compact constant 1.84 in SI so that Q = 1.84 (L − 0.1 n H) H^(3/2).

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis confirms m^3/s when L and H are in meters. Field calibrations show good agreement within the method’s range.

Why Other Options Are Wrong:

Options with H^2 or H^(5/2) do not match the sharp-crested exponent. Using n H instead of 0.1 n H overcorrects end contractions. Expressions with Cd 2 g are incomplete without integration and constants consolidation.

Common Pitfalls:

Measuring H too close to the crest, inadequate aeration of the nappe, or miscounting end contractions.

Final Answer:

Q = 1.84 (L − 0.1 n H) H^(3/2)